Learning path

Each declaration appears after its prerequisites.

Arxiv.«0805.3777».IsGenericRank IsGenericRank ds r says that r is the generic rank of complex tensors of format ds: the tensors of rank r contain a dense open set
Arxiv.«0805.3777».IsMaxRank IsMaxRank ds r says that r is the maximal rank of a complex tensor of format ds: every such tensor has rank at most r, and some tensor has rank r
Arxiv.«0805.3777».cprank_eq_zero_iff A tensor has rank 0 exactly when it is zero
Arxiv.«0805.3777».IsMaxRank.le_prod The maximal rank of a tensor of format ds is at most ds.prod
Arxiv.«0805.3777».IsGenericRank.unique There is at most one generic rank
Arxiv.«0805.3777».IsGenericRank.le_of_isMaxRank The generic rank is at most the maximal rank [Fri12, (4.3)]
Arxiv.«0805.3777».isMaxRank_one The maximal rank of a $1 \times 1 \times 1$ tensor is 1
Arxiv.«0805.3777».isGenericRank_matrix The generic rank of an $m \times n$ matrix is $\min(m, n)$: the 2-tensor case
Arxiv.«0805.3777».isGenericRank_of_gt_critical Above the critical range, a generic tensor of format $(m_1, m_2, m_3)$ has rank $\min(m_3, m_1m_2)$ [Fri12, (5.1)]
Arxiv.«0805.3777».isGenericRank_three_even The generic rank of a $3 \times 2p \times 2p$ tensor is $\lceil 12p^2 / (4p + 1) \rceil$, the value $r_0$ predicted by a dimension count [Fri12, (5.3)]
Arxiv.«0805.3777».isGenericRank_three_odd Documented declaration: isGenericRank_three_odd
Arxiv.«0805.3777».isGenericRank_four The generic rank of a $4 \times m \times m$ tensor is $\lceil 4m^2 / (2m + 2) \rceil$ [Fri12, (5.7)]
Arxiv.«0805.3777».isGenericRank_cube For $n \neq 3$ the generic rank of an $n \times n \times n$ tensor is $\lceil n^3 / (3n - 2) \rceil$ [Fri12, (5.8)]
Arxiv.«0805.3777».isGenericRank_of_le_critical Friedland's conjecture
Arxiv.«0805.3777».isMaxRank_two The maximal rank of a $2 \times m \times n$ tensor is $m + \min(m, \lfloor n/2 \rfloor)$ [Fri12, (5.13)]; equivalently [BFZ24, (4.9)]
Arxiv.«0805.3777».cprank_le_of_three_slices Every $n \times n \times 3$ tensor has rank at most $2n - 1$ [SMS10, Theorem 4.5]
Arxiv.«0805.3777».isMaxRank_three The maximal rank of a $3 \times 3 \times 3$ tensor is 5
Arxiv.«0805.3777».isMaxRank_of_le_min For $u \leq \min(4, m)$ the maximal rank of an $m \times n \times (mn - u)$ tensor is $mn - \lceil u/2 \rceil$ [BFZ24, (4.24)]
Arxiv.«0805.3777».isMaxRank_three_three_five_bounds The maximal rank of a $3 \times 3 \times 5$ tensor is 6 or 7 [BFZ24, (4.20)]
Arxiv.«0805.3777».isMaxRank_le_two_mul_isGenericRank The maximal rank is at most twice the generic rank [BT15, Theorem 1]
Arxiv.«0805.3777».isMaxRank_cube Open problem
Arxiv.«0911.2077».arxiv.id0911_2077.conjecture6_3 Empirical evidence seems to suggest that Slud's bound does not hold for all $p$, and in fact, as $n\to\infty$, the maximal permissible $p$ shrinks to $\frac{1}{2}$
Arxiv.«0912.2382».k The curling number Let $S$ be a finite nonempty sequence of integers
Arxiv.«0912.2382».S One starts with any initial sequence of integers $S₀$, and extends it by repeatedly appending the curling number of the current sequence
Arxiv.«0912.2382».curling_number_conjecture The sequence will eventually reach $1$
AtiyahSutcliffe.Point A point of Euclidean three-space
AtiyahSutcliffe.directionLift A deterministic projective lift of a direction in $\mathbb{R}^3$ to a pair of complex numbers
AtiyahSutcliffe.linearFactor The homogeneous linear factor determined by a lifted direction
AtiyahSutcliffe.pointPolynomial The polynomial associated to point i in a finite configuration x
AtiyahSutcliffe.conjecture_one Atiyah–Sutcliffe Conjecture 1, stated as Conjecture 1.1 in Mazur–Petrenko: the configuration polynomials are linearly independent
CunninghamChain.firstKindTerm The nth term generated from p by the first-kind recurrence q ↦ 2q + 1
CunninghamChain.secondKindTerm The nth term generated from p by the second-kind recurrence q ↦ 2q - 1
CunninghamChain.IsFirstKindChainOfLength p starts a first-kind Cunningham chain of exact positive length k
CunninghamChain.IsSecondKindChainOfLength p starts a second-kind Cunningham chain of exact positive length k
CunninghamChain.two_starts_firstKind_length_five Sanity-check: 2 starts a first-kind chain of length 5 (2, 5, 11, 23, 47 are prime; 2·47+1 = 95 = 5·19 is composite)
CunninghamChain.seven_starts_secondKind_length_two Sanity-check: 7 starts a second-kind chain of length 2 (7, 13 are prime; 2·13-1 = 25 = 5² is composite)
CunninghamChain.infinitely_many_firstKind_chains Jones's conjecture (first kind): for every positive integer $k$, there are infinitely many primes $p$ that start a first-kind Cunningham chain of exactly length $k$
CunninghamChain.infinitely_many_secondKind_chains Jones's conjecture (second kind): for every positive integer $k$, there are infinitely many primes $p$ that start a second-kind Cunningham chain of exactly length $k$
Arxiv.«1308.0994».Formula Formula is the inductive type of propositional modal formulas: * Atom n is a propositional variable indexed by n
Arxiv.«1308.0994».Conj Conj α β is the conjunction α ∧ β
Arxiv.«1308.0994».t t φ is the Boxdot translation of a formula φ
Arxiv.«1308.0994».Formula.subst Formula.subst σ φ is the uniform substitution of σ n for each propositional variable Atom n in φ
Arxiv.«1308.0994».KProof KProof Γ φ is the usual Hilbert‐style proof relation for the minimal normal modal logic K, with assumptions drawn from Γ
Arxiv.«1308.0994».KTProof KTProof Γ φ denotes that φ is provable from the premises Γ in the normal modal logic KT (also called T)
Arxiv.«1308.0994».KTExtendsK If KProof Γ φ, then KTProof Γ φ
Arxiv.«1308.0994».KProof.subst Provability in K is closed under uniform substitution
Arxiv.«1308.0994».KTProof.subst Provability in KT is closed under uniform substitution
Arxiv.«1308.0994».NormalModalLogic A “normal modal logic” L is any Set Formula such that: 1
Arxiv.«1308.0994».KT KT is the specific normal modal logic whose theorems are exactly those provable in KTProof from the empty context
Arxiv.«1308.0994».BoxdotConjecture The formal proof was done by Mashu Noguchi et al
Arxiv.«1601.03081».IsCrystalWithComponents An odd number $n$ is called a crystal if $n = ab$, with $a, b > 1$ and $B(a, b) ∈ ℕ$, where $B(a, b) := ((a + b)^2 + (a b + 1)^2) / (2 (a + 1) (b + 1))$
Arxiv.«1601.03081».crystals_components_unique Documented declaration: crystals_components_unique
Arxiv.«1609.08688».lt₂ Let $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ be two triples of integers
Arxiv.«1609.08688».lt₂_example_1 For example, $(3, 3, 9) <_2 (5, 6, 1)$
Arxiv.«1609.08688».lt₂_example_2 $(5, 6, 1) <_2 (7, 7, 7)$
Arxiv.«1609.08688».lt₂_example_3 $(7, 7, 7) <_2 (7, 8, 9)$
Arxiv.«1609.08688».not_lt₂_example but $(1, 2, 3)$ is not $2$-less than $(1, 2, 4)$
Arxiv.«1609.08688».not_trans_lt₂_nat The $2$-less relation is not transitive on the naturals
Arxiv.«1609.08688».IsIncreasing₂ Since the $2$-less relation is not transitive, we make a further definition to specify transivity
Arxiv.«1609.08688».maximalLength Let $F(n)$ be the maximal length of a $2$-increasing sequence of triples with each coordinate belong to $[n]$ ($= \{1, 2, ..., n\}$)
Arxiv.«1609.08688».exists_pair_of_mem_Icc In a set of more than $n^2$ triples with coordinates from $\{1, ..., n\}$ we must have two triples that are equal in their first two coordinates
Arxiv.«1609.08688».maximalLength_le For all $n$ we have $F(n) \leq n^2$
Arxiv.«1609.08688».maximalLength_ge_of_isSquare Moreover, whenever $n$ is a perfect square we have $F(n) \geq n^{3/2}$
Arxiv.«1609.08688».IsComparable₂ Two triples $t_1$ and $t_2$ are $2$-comparable if one of them is $2$-less than the other
Arxiv.«1609.08688».IsComparableSet₂ A set of triples is $2$-comparable if any two distinct members of it are $2$-comparable
Arxiv.«1609.08688».maximalLength_le_isBigO $F(n) \leq n^2 / \exp(\Omega(\log^*(n)))$, i.e. there is a constant $c > 0$ such that $F(n) \leq n^2 / \exp(c \log^*(n))$ for all sufficiently large $n$
Arxiv.«1609.08688».tripleProduct We define the product of two triples $(a, b, c)$ and $(d, e, f)$ by $((a, d), (b, e), (c, f))$, where the pairs are ordered lexicographically
Arxiv.«1609.08688».sequenceProduct We define the product
Arxiv.«1609.08688».isIncreasing₂_sequenceProduct_example The product of two $2$-increasing sequences is $2$-increasing; this is the point of ordering the pairs lexicographically
Arxiv.«1609.08688».maximalLength_pow Suppose that for some $n$ we have $F(n) = n ^ {\alpha}$
Arxiv.«1609.08688».maximalLength_pow_eventually Moreover, for every $\beta < \alpha$ and all sufficiently large $m$ we have $F(m) \geq m^{\beta}$
Arxiv.«1609.08688».maximalLength_le_strong $F(n) \leq n^{3/2}$
Arxiv.«2001.02665».ringel_conjecture_large For all sufficiently large $n$, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of any tree $T$ with $n$ edges
Arxiv.«2001.02665».kotzig_conjecture_large For all sufficiently large $n$, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of any tree $T$ with $n$ edges via cyclic shifts of a single embedding
Arxiv.«2104.00502».aperiodicAutocorrelation The aperiodic autocorrelation $C_k(a)=\sum_i a_i a_{i+k}$
Arxiv.«2104.00502».IsBarkerSequence A Barker sequence has $\pm 1$ entries and nontrivial autocorrelations of magnitude at most one
Arxiv.«2104.00502».barker_conjecture Every Barker sequence has length at most $13$
Arxiv.«2104.00502».isBarkerSequence_length_thirteen The known sequence $(1,1,1,1,1,-1,-1,1,1,-1,1,-1,1)$ of length $13$ is a Barker sequence
Arxiv.«2104.00502».isBarkerSequence_length_four The sequence $(1,1,1,-1)$ is a Barker sequence
Arxiv.«2104.00502».not_isBarkerSequence_constant_four The constant sequence $(1,1,1,1)$ is not a Barker sequence
Arxiv.«2107.00295».independentDominationEven Conjecture 1.6 (Even case)
Arxiv.«2107.00295».independentDominationOdd Conjecture 1.6 (Odd case)
Arxiv.«2107.12475».CollatzLike For $n > 8$, $2^n$ is not the the sum of distinct powers of $3$
Arxiv.«2107.12475».two_not_in_digits_three_pow_eight For $n = 8$, $2$ is not contained in the base $3$ digits of $n$
Arxiv.«2208.14736».IsCancellative A finitely generated k-algebra A is cancellative if for all finitely generated k algebras B such that B[X] ≅ₖ A[X] we have B ≅ₖ A
Arxiv.«2208.14736».zariski_cancellation_problem The Zariski Cancellation Problem: every polynomial ring over a field k of characteristic 0 is cancellative
Arxiv.«2208.14736».zariski_cancellation_problem.variants.dim_one The single variable polynomial ring k[X] is cancellative in any characteristic
Arxiv.«2208.14736».zariski_cancellation_problem.variants.dim_two The two variable polynomial ring k[X] is cancellative in any characteristic
Arxiv.«2208.14736».zariski_cancellation_problem.variants.false_pos_card The positive characteristic case of the Zariski Cancellation Problem is false in dimension 3
NowhereDenseSpectralSet.exists_nowhereDense_spectralSet [KLM2023, Problem 7.1] asks whether a bounded, measurable, nowhere dense subset $\Omega \subset \mathbb{R}^d$ of positive measure can be spectral
SpectralSetProduct.productSet The product $A \times B$, represented by consecutive coordinate blocks
SpectralSetProduct.spectralProductImpliesRightSpectral Spectrality of a product with an n-dimensional convex body forces spectrality of its bounded, measurable m-dimensional right factor
SpectralSetProduct.isSpectral_right_of_product_one_dimensional [KLM2023, Problem 7.2; GL16] For a one-dimensional convex body $A$ and a bounded, measurable set $B$, if $A \times B$ is spectral, then $B$ is spectral
SpectralSetProduct.isSpectral_right_of_product_two_dimensional [KLM2023, Problem 7.2; GL20] For a two-dimensional convex body $A$ and a bounded, measurable set $B$, if $A \times B$ is spectral, then $B$ is spectral
SpectralSetProduct.isSpectral_right_of_product_of_convexBody [KLM2023, Problem 7.2] For every positive dimension $n$, a convex body $A$ and a bounded, measurable set $B$, must spectrality of $A \times B$ imply spectrality of $B$?
Arxiv.id2303_01089.Tn The map $T_n$ sends $x$ to $nx \bmod 1$ on the additive circle
Arxiv.id2303_01089.IsTnInvariant A set $F$ is $T_n$-invariant if $T_n(F) \subseteq F$
Arxiv.id2303_01089.MeasureTheory.IsAtom A set $A$ is an atom if it has positive measure and for all $B \subseteq A$ measurable, either $\mu(B) = 0$ or $\mu(B) = \mu(A)$
Arxiv.id2303_01089.MeasureTheory.IsAtomLess A measure is atomless if it has no atoms
Arxiv.id2303_01089.conjecture_1_3 Conjecture 1.3 (the $\times p, \times q$ conjecture): the only atomless Borel probability measure on $\mathbb{T}$ which is both $T_p$- and $T_q$-invariant is the Lebesgue measure
Arxiv.id2303_01089.conjecture_1_4 Conjecture 1.4: if $\mu$ is an atomless $T_p$-invariant Borel probability measure on $\mathbb{T}$, then $T_{q^n}\mu$ converges weak-star to Lebesgue measure
CirculantHadamard.orderFourGenerator A generator for a circulant Hadamard matrix of order four
CirculantHadamard.orderFourGenerator_isHadamard The order-four generator gives a circulant Hadamard matrix
CirculantHadamard.circulant_hadamard_conjecture Every circulant Hadamard matrix has order at most four
Arxiv.«2501.03234».S' Define the sum $$S'(h, k) := \sum_{j=1}^{k-1}(-1)^{j + 1 + \lfloor \frac{hj}{k}\rfloor}.$$
Arxiv.«2501.03234».S Define the sum $$S(k) := \sum_{h=1}^{k-1}S'(h, k)$$
Arxiv.«2501.03234».S_fst_10 Note that in Table 1 in https://arxiv.org/abs/2501.03234v1, there seems to be an error: 11 appears twice
Arxiv.«2501.03234».conjecture_1_1 Conjecture 1.1: For any odd prime $k$, the sum associated with the classical theta function $θ_3$, $S(k)$ is positive
Arxiv.«2501.03234».conjecture_4_1 Conjecture 4.1: For any prime $k$ larger than $5$, $S(k) > k$
Arxiv.«2501.03234».conjecture_4_2 Conjecture 4.2: For any prime $k$ larger than $233$, $S(k) > 2k$
Arxiv.«2501.03234».conjecture_4_3 Conjecture 4.3: For any prime $k$ larger than $3119$, $S(k) > 3k$
Arxiv.«2501.03234».conjecture_4_4 Conjecture 4.4: Given a natural number $n ∈ ℕ$, for all large enough odd prime $k$ (depending on $n$), $nk < S(k)$
Arxiv.«2501.03234».conjecture_4_4_def_0 Conjecture 1.1 → Conjecture 4.4: If conjecture 1.1 holds true, then this implies a special case of conjecture 4.4 where $n = 0$
Arxiv.«2501.03234».conjecture_4_4_def_1 Conjecture 4.1 → Conjecture 4.4: If conjecture 4.1 holds true, then this implies a special case of conjecture 4.4 where $n = 1$
Arxiv.«2501.03234».conjecture_4_4_def_2 Conjecture 4.2 → Conjecture 4.4: If conjecture 4.2 holds true, then this implies a special case of conjecture 4.4 for $n = 2$
Arxiv.«2501.03234».conjecture_4_4_def_3 Conjecture 4.3 → Conjecture 4.4: If conjecture 4.3 holds true, then a special case of conjecture 4.4 for $n = 3$ is obtained, and the lower bound is $3119$
Margulis.conjecture_1_1 Let D be the diagonal group of SL_n(ℝ) where n ≥ 3
Margulis.polyToLaurent The natural inclusion F[t] →+* F((t⁻¹))
Margulis.huang_shi_theorem_1_2 Huang–Shi, Theorem 1.2 Let F be a finite field of characteristic p ∈ {3, 5, 7, 11}, and set K = F((t⁻¹)), A = F[t]
Arxiv.«2507.17780».tx_graffiti_conjecture_1 TxGraffiti Conjecture 1: for every nontrivial connected graph $G$ with $\Delta(G) \ge 2$, $$\alpha(G) \ge \frac{a(G) + R(G)}{\Delta(G)}.$$ This conjecture is true
Arxiv.«2507.17780».zeroForcingStep One step of the zero forcing colour-change rule: add to $S$ every vertex $w$ that is the unique neighbour of some $v \in S$ outside $S$
Arxiv.«2507.17780».zeroForcingClosure The zero forcing closure: iterate zeroForcingStep until it stabilises
Arxiv.«2507.17780».IsZeroForcingSet A set $S$ is a zero forcing set of $G$ if its zero forcing closure is all of $V$
Arxiv.«2507.17780».zeroForcingNumber The zero forcing number of a finite simple graph: the minimum cardinality of a zero forcing set
Arxiv.«2507.17780».tx_graffiti_conjecture_2 TxGraffiti Conjecture 2: for every connected graph $G$ with $\Delta(G) \le 3$ and $G \ne K_4$, $$Z(G) \le \alpha(G) + 1.$$ This conjecture is false
Arxiv.«2507.17780».tx_graffiti_conjecture_3 TxGraffiti Conjecture 3: for every $r$-regular graph $G$ ($r \ge 1$), $$i(G) \le \mu^*(G).$$ This conjecture is open
Arxiv.«2507.17780».friendshipF4 The friendship graph $F_4$: a hub vertex $0$ joined to four triangles, whose rims are the pairs $\{1,2\}, \{3,4\}, \{5,6\}, \{7,8\}$
Arxiv.«2507.17780».tx_graffiti_conjecture_4 TxGraffiti Conjecture 4: for every nontrivial connected graph $G$, the saturation number is at most the harmonic index, $\mu^*(G) \le H(G)$
Arxiv.«2602.05192».finiteAdditiveConvolution Define
Arxiv.«2602.05192».Φ For a monic polynomial
Arxiv.«2602.05192».FourProp A predicate that holds if $p(x)$ and $q(x)$ are monic real-rooted polynomials of degree $n$, then $$\frac{1}{\Phi_n(p\boxplus_n q)} \ge \frac{1}{\Phi_n(p)}+\frac{1}{\Phi_n(q)}?$$
Arxiv.«2602.05192».four_2 Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of degree $2$, then $$\frac{1}{\Phi_2(p\boxplus_n q)} \ge \frac{1}{\Phi_2(p)}+\frac{1}{\Phi_2(q)}?$$
Arxiv.«2602.05192».four_3 Is it true that if $p(x)$ and $q(x)$ are monic real-rooted polynomials of degree $3$, then $$\frac{1}{\Phi_3(p\boxplus_n q)} \ge \frac{1}{\Phi_3(p)}+\frac{1}{\Phi_3(q)}?$$
Arxiv.«2602.05192».IsEpsilonLight For a graph $G = (V, E)$, let $G_S = (V, E(S,S))$ denote the graph with the same vertex set, but only the edges between vertices in $S$
Arxiv.«2604.08040».cyc The number of cyclic subgroups of a finite group G
Arxiv.«2604.08040».numPrimeFactors The number of distinct prime divisors of the order of G
Arxiv.«2604.08040».nilpotent_of_cyc_lt Theorem 3.1
Arxiv.«2604.08040».solvable_of_cyc_lt_two_pow_succ Theorem 4.2
Arxiv.«2604.08040».card_alternatingGroup_fin_five The alternating group on five letters has order $60 = 2^2 \cdot 3 \cdot 5$
Arxiv.«2604.08040».cyc_alternatingGroup_five The alternating group on five letters has exactly $32$ cyclic subgroups: the trivial one, $15$ of order $2$, $10$ of order $3$ and $6$ of order $5$
Arxiv.«2604.08040».not_cyc_alternatingGroup_five_lt $A_5$ is the sharpness witness: it has $\pi(A_5) = 3$ and $\mathrm{cyc}(A_5) = 32 = 2^{3+2}$, so it misses the strict inequality $\mathrm{cyc}(G) < 2^{t+2}$ by exactly $1$
Arxiv.«2604.08040».cyc_punit The trivial group has exactly one cyclic subgroup (itself)
Arxiv.«2604.08040».numPrimeFactors_trivial The trivial group has zero prime divisors
Arxiv.«2605.02731».dean_conjecture Conjecture 1.1 (Dean, 1988)
Arxiv.«2605.02731».dean_conjecture.variants.five The case $k = 5$
Arxiv.«2605.02731».dean_conjecture.variants.three The case $k = 3$, proved by Chen and Saito [ChSa94]
Arxiv.«2605.02731».dean_conjecture.variants.four The case $k = 4$, proved by Dean, Lesniak and Saito [DeLeSa93]
Arxiv.«2605.02731».dean_conjecture.variants.six_le The cases $k \geq 6$, proved by Luo, Ma and Zhao [LuMaZh26]
Arxiv.«2605.02731».minDegree_bot_eq_zero A graph with no edges has minimum degree 0, so the hypothesis of the conjecture rules it out
Arxiv.«2605.02731».minDegree_top_fin_four The complete graph on four vertices has minimum degree 3, so it is one of the graphs the case k = 3 applies to
Arxiv.«2605.12342».signDiffHom The group homomorphism $(\sigma_1, \sigma_2) \mapsto \mathrm{sgn}(\sigma_1) \cdot \mathrm{sgn}(\sigma_2)^{-1}$ from $\mathrm{S}_m \times \mathrm{S}_n$ to $\{+1, -1\}$
Arxiv.«2605.12342».gammaSubgroup Documented declaration: gammaSubgroup
Arxiv.«2605.12342».conjecture_1.variants.rank_2_2 It is known that $\Gamma_{2 \oplus 2} \cong C_2$ has rank $1$: it is cyclic, generated by a single element
Arxiv.«2605.12342».conjecture_1.variants.rank_3_3 It is known that $\Gamma_{3 \oplus 3}$ has rank $3$: it is not $2$-generated
Arxiv.«2605.12342».conjecture_1.variants.rank_4_3 It is known that $\Gamma_{4 \oplus 3}$ has rank $3$: it is not $2$-generated
Arxiv.«2605.12342».conjecture_1.variants.rank_4_4 It is known that $\Gamma_{4 \oplus 4}$ has rank $3$: it is not $2$-generated
Arxiv.«2606.03696».offWalk The set of vertices that a walk does not touch
Arxiv.«2606.03696».bondy_conjecture Conjecture 1 (Bondy, 1980)
Arxiv.«2606.03696».bondy_conjecture.variants.one The case $k = 1$ is Dirac's theorem
Arxiv.«2606.03696».bondy_conjecture.variants.two The case $k = 2$ is the theorem of Nash-Williams
Arxiv.«2606.03696».bondy_conjecture.variants.three The case $k = 3$ is proved
Arxiv.«2606.03696».bondy_conjecture.variants.large Theorem 1.1 (Ma-Ning-Zhao, 2026)
Arxiv.«2607.03582».supportFunction The support function $h_K(u) = \sup_{x \in K} \langle x, u\rangle$
Arxiv.«2607.03582».lpSum The Firey $L_p$-sum $K \oplus_p L$, equation (4) of the source: the body whose support function is $(h_K^p + h_L^p)^{1/p}$
Arxiv.«2607.03582».HasCentreOfSymmetry $K$ has a centre of symmetry: some $c$ with $K$ invariant under reflection in $c$
Arxiv.«2607.03582».rsConstant The constant of Corollary 29, $\frac{2\Gamma(1+1/q)^2}{\Gamma(1+2/q)} + 2$
Arxiv.«2607.03582».IsParallelogramAtOrigin $K$ is a parallelogram with a vertex at the origin: the convex hull of $0$, $a$, $b$ and $a + b$ for linearly independent $a$, $b$
Arxiv.«2607.03582».volume_lpSum_le Corollary 29 (Fradelizi-Manui-Meyer-Ndiaye, 2026)
Arxiv.«2607.03582».volume_lpSum_eq_of_isParallelogramAtOrigin Parallelograms with a vertex at the origin attain the bound, which is the sharpness half of Corollary 29
Arxiv.«2607.03582».rsConstant_two At $p = 2$ the conjugate is also $2$, and $\Gamma(3/2)^2 / \Gamma(2) = \pi/4$, so the constant is $\pi/2 + 2$
Arxiv.«2607.05349».similarityMatrix The similarity matrix $Z(t)_{ij} = e^{-t \cdot d(x_i, x_j)}$ of a finite metric space
Arxiv.«2607.05349».distanceMatrix The distance matrix $D_{ij} = d(x_i, x_j)$ of a finite metric space
Arxiv.«2607.05349».weighting The weighting $\vec{w}(t) = Z(t)^{-1}\mathbf{1}$ at scale t
Arxiv.«2607.05349».HasMicroscopicWeighting X admits a microscopic weighting when $\vec{w}(t)$ converges as $t \to 0^+$
Arxiv.«2607.05349».IsGauging g is a gauging for M with concentration c: it sums to 1 and M g is the constant vector c
Arxiv.«2607.05349».concentration_unique All gaugings of a symmetric matrix share one concentration, so $\mathrm{con}$ is an attribute of the matrix
Arxiv.«2607.05349».HasFiniteConcentration M has finite concentration, written $\mathrm{con}(M) \neq \infty$ in the source: some gauging for M exists
Arxiv.«2607.05349».hasFiniteConcentration_of_hasMicroscopicWeighting One direction is known: a microscopic weighting implies finite concentration (Theorem 3.1(3))
Arxiv.«2607.05349».hasMicroscopicWeighting_iff_of_isUnit Theorem 3.8: the conjecture is known when the distance matrix is invertible, and the microscopic weighting is then the unique gauging
Arxiv.«2607.05739».gaussProd $Z_n = \prod_{k=1}^n (1 + ik)$, in the Gaussian integers
Arxiv.«2607.05739».A $A_n = \operatorname{Re} Z_n$
Arxiv.«2607.05739».B $B_n = \operatorname{Im} Z_n$
Arxiv.«2607.05739».x $x_n = \tan\left(\sum_{k=1}^n \arctan k\right) = B_n / A_n$
Arxiv.«2607.05739».IsIntegerValue $x_n$ takes an integer value
Arxiv.«2607.05739».tan_arctan_sum_not_integer Conjecture (Amdeberhan-Medina-Moll, 2008)
Arxiv.«2607.05739».x_succ $x_n$ satisfies $x_1 = 1$ and $x_n = \dfrac{x_{n-1} + n}{1 - n x_{n-1}}$, which is the tangent addition formula
Arxiv.«2607.05739».x_eq_tan_sum_arctan $x_n$ is the tangent of the partial sum of arctangents it is named for
Arxiv.«2607.05739».omega $\omega_n = A_n^2 + B_n^2 = \prod_{k=1}^n (1 + k^2)$, the norm of $Z_n$
Arxiv.«2607.05739».kernel The squarefree kernel $K_n$ of $\omega_n$: the product of the primes dividing it to an odd power, so 1 when there are none
Arxiv.«2607.05739».angleSum $a_n = \sum_{k=1}^n \arctan(1/k)$
Arxiv.«2607.05739».exceptional The exceptional set $E = \{n \geq 5 : |x_n| > n/2 + 1\}$
Arxiv.«2607.05739».isIntegerValue_of_le_four The four integer values the conjecture leaves out
Arxiv.«2607.05739».x_values $x_1 = 1$, $x_2 = -3$, $x_3 = 0$, $x_4 = 4$
Arxiv.«2607.05739».not_isIntegerValue_of_mem_Icc_five_ten The conjecture holds at the first few values it covers
Arxiv.«2607.06396».IsCycleCover C covers G: every edge of G lies on at least one cycle of C
Arxiv.«2607.06396».totalLength The total length of a family of cycles
Arxiv.«2607.06396».alon_tarsi_short_cycle_cover Conjecture 4 (Alon-Tarsi, 1985)
Arxiv.«2607.06396».exists_cover_of_edgeFinset_eq_empty The empty cover works when there are no edges, so the bound is attained with room to spare on edgeless graphs
Arxiv.«2607.06396».exists_cover_of_isAcyclic Acyclic bridgeless graphs satisfy the conjecture, with the empty cover
Arxiv.«2607.06396».three_le_length A cycle has at least three edges, so any cover of a graph with an edge has total length at least three
Arxiv.«2607.08366».IsValidMod A is valid mod N when the all-ones multiset is the only multiset of size #A drawn from A whose sum matches ∑ a ∈ A, a
Arxiv.«2607.08366».minModulus The candidate least modulus admitting a valid set of n residues, conjecturally $2^n - 2^{\lfloor\log_2 n\rfloor}$
Arxiv.«2607.08366».not_isValidMod_iff Invalidity has a multiplicity witness that differs from one at an element of the set
Arxiv.«2607.08366».minModulus_pos The candidate modulus is positive for every positive set size
Arxiv.«2607.08366».minModulus_lt_two_pow The candidate modulus is strictly below the full power of two
Arxiv.«2607.08366».min_modulus Conjecture 1 (Fonollosa, 2026)
Arxiv.«2607.08366».one_sum_eq The all-ones multiset always has the right size and the right sum, so IsValidMod is a uniqueness statement rather than an existence one
Arxiv.«2607.08366».isValidMod_of_subsingleton Documented declaration: isValidMod_of_subsingleton
Arxiv.«math.0110202».banach_mazur_rotation_problem Documented declaration: banach_mazur_rotation_problem
Arxiv.«math.0110202».banach_mazur_rotation_problem.finite_dimensional Every finite-dimensional real normed space whose isometry group acts transitively on the unit sphere is Euclidean
HopfAlgebraCohomology.cohomologyRing_finiteType Finite generation conjecture (Etingof–Ostrik, Conjecture 2.18, algebra part)
HopfAlgebraCohomology.cohomology_finite Finite generation conjecture (Etingof–Ostrik, Conjecture 2.18, module part)
HopfAlgebraCohomology.cohomologyRing_finiteType_of_isCocomm Friedlander–Suslin
HopfAlgebraCohomology.cohomology_finite_of_isCocomm Friedlander–Suslin
HopfAlgebraCohomology.cohomologyRing_finiteType_monoidAlgebra Evens–Venkov
HopfAlgebraCohomology.cohomology_trivialModuleCat Sanity check: taking the coefficient module M of the module statement to be k itself gives back the cohomology ring
HopfAlgebraCohomology.cohomologyRing_nontrivial Sanity check: the cohomology ring is not the zero ring, over which every finiteness statement would hold trivially
Arxiv.«math.0608009».IsPoissonEndomorphism Documented declaration: IsPoissonEndomorphism
Arxiv.«math.0608009».PoissonConjectureFor The Poisson Conjecture in dimension n ($PC_n$ in the notation of [AvdE07]): every endomorphism of the n-th canonical Poisson algebra over K is an automorphism
Arxiv.«math.0608009».poisson_conjecture Documented declaration: poisson_conjecture
Arxiv.«math.0608009».poisson_conjecture.variants.dimension_one The Poisson Conjecture in dimension $1$ ($PC_1$) is open
Arxiv.«math.0608009».poisson_conjecture.variants.dimension_two The Poisson Conjecture is false in dimension $2$ ($PC_2$)
Arxiv.«math.0608009».poisson_conjecture.variants.dimension_three The Poisson Conjecture is false in dimension
Arxiv.«math.0608009».poisson_conjecture.variants.dimension_ge_two The Poisson Conjecture is false in every dimension $n ≥ 2$, by padding the dimension $2$ counterexample of [Lon26] with identity canonical pairs
Arxiv.«math.0608009».poisson_conjecture.variants.jacobian_implication The Jacobian conjecture in dimension $2n$ implies the Poisson conjecture in dimension n ([AvdE07], Theorem 1 and Theorem 7)
Arxiv.«math.0608009».isPoissonEndomorphism_id The identity is a Poisson endomorphism
Arxiv.«math.0608009».poissonBracket_X_pairing The canonical bracket pairs each position variable with its momentum variable
Arxiv.«math.0608009».poissonConjectureFor_zero The Poisson conjecture holds trivially in dimension $0$, where the Poisson algebra is K itself and the identity is the only endomorphism
Arxiv.«math.0703548».matrixHattoriStallingsTraceAt The value at the conjugacy class of g of the universal trace of a matrix over R[G]
Arxiv.«math.0703548».projective_module_isStablyFree Vanishing of the reduced projective class group for integral group rings
Arxiv.«math.0703548».bass_conjecture_integral_domain Bass conjecture for commutative integral domains, in idempotent-matrix form
BorweinSineSeries.borwein_sine_series Does the series $$ \sum_{n=1}^{\infty} \frac{\left(\frac{2}{3} + \frac{1}{3}\sin n\right)^n}{n} $$ converge?
Bugeaud01.problem_10_1 Problem 10.1
Bugeaud01.problem_10_2 Problem 10.2
Bugeaud01.problem_10_3 Problem 10.3
Bugeaud01.waldschmidt Waldschmidt [Wal03] conjectured that a stronger result holds, namely that there exists a positive real number~$c$ such that $\lVert e^n \rVert > n^{-c}$ for every~$n \ge 2$
Bugeaud01.problem_10_3_of_waldschmidt Waldschmidt's conjecture is stronger than Mahler's: since $\log n \le n$ for $n \ge 1$, the polynomial lower bound $n^{-c}$ dominates the exponential lower bound $e^{-cn}$
Bugeaud32.adamczewski_faverjon Adamczewski–Faverjon [AF20, Conjecture A.3], proved there from Part (i) of Theorem 1.1
Bugeaud32.problem_10_32.variants.strong Problem 10.32, strong form
Bugeaud32.problem_10_32 Problem 10.32
Bugeaud04.Spectrum The spectrum of a sequence
Bugeaud05.problem_10_5_of_moreover The "moreover" form of Problem 10.5 implies the first part: applying the cluster-point density to the subinterval
Bugeaud53.problem_10_53 Problem 10.53
Bugeaud53.problem_10_53.variants.all_bases Problem 10.53, in the stronger form that asks for every base at once: there is an irrational real number
Bugeaud53.problem_10_53.variants.uncountable There are uncountably many numbers as in Problem 10.53
Bugeaud53.problem_10_53.variants.temur Temur [Tem26], Theorem 1
Bugeaud06.pollington_de_mathan The Pollington–de Mathan theorem [Pol79][Mat80]
Bugeaud06.furstenberg_two_three Furstenberg's theorem [Fur67] (the $\times 2, \times 3$ case)
Bugeaud06.boshernitzan Boshernitzan's theorem [Bos94]
Bugeaud06.countingFn The counting function $\pi_A(N) = \#(A \cap [0, N])$ of the range $A$ of m
Bugeaud06.mSeq The sequence defined by $m_0 = 2$ and $m_{n+1} = \lceil m_n (1 + 1/\log n) \rceil$
Bugeaud06.IsGenuinelySublacunary The sequence $m$ eventually grows at least geometrically with a logarithmic correction
Bugeaud06.example_isGenuineSublacunary Documented declaration: example_isGenuineSublacunary
Bugeaud06.isGenuinelySublacunary_of_isLacunary A ratio floor is weaker than lacunarity: a lacunary sequence has ratios bounded below by a constant $> 1$, which dominates $1 + c/\log n$
Bugeaud06.HasIntermediateGrowth The sequence $m$ eventually grows at least as fast as $\exp(n^{\alpha})$, i.e., super-exponential growth when $\alpha > 1$, and stretched-exponential when $0 < \alpha < 1$
Bugeaud06.example_hasIntermediateGrowth mSeq has intermediate (subexponential but super-polynomial) growth: for every 0 < α < 1 its terms eventually dominate $\exp(n^\alpha)$
Bugeaud06.not_isLacunary_of_dense A sequence that is dense modulo one for every irrational
Bugeaud06.problem_10_6_variant_1 Problem 10.6
Bugeaud06.problem_10_6_variant_2 Problem 10.6, intermediate-growth variant
Bugeaud61.cantorPoint The point of $C(\alpha)$ with digit sequence $\varepsilon$, that is $(\alpha - 1) \sum_{k \ge 1} \varepsilon_k \alpha^{-k}$
Bugeaud61.pisotCantorSet The set $C(\alpha)$ of Problem 10.61
Bugeaud61.routeAExponent The Route A exponent
Bugeaud61.problem_10_61 Problem 10.61
Bugeaud61.problem_10_61.variants.almost_every Documented declaration: almost_every
Bugeaud61.zero_mem_pisotCantorSet Sanity check: $0 \in C(\alpha)$, the point with all digits zero
Bugeaud61.one_mem_pisotCantorSet Sanity check: $1 \in C(\alpha)$ for $\alpha > 1$, the point with all digits one, since $\sum_{k \ge 1} \alpha^{-k} = 1 / (\alpha - 1)$
Bugeaud07.problem_10_7 Problem 10.7
Bugeaud08.problem_10_8 Problem 10.8 ($p$-adic Littlewood conjecture)
Bugeaud08.problem_10_8.variants.quadratic The quadratic case of Problem 10.8
Bugeaud08.problem_10_8.variants.exceptional_set_dimH_zero The exceptional set for Problem 10.8 has Hausdorff dimension zero (Einsiedler and Kleinbock [EK07]
Bugeaud09.problem_10_9 Problem 10.9
Bugeaud09.not_isZNumber_one Sanity check: 1 is not a Z-number, since for n = 1 we have $\{1 \cdot (3/2)^1\} = \{3/2\} = 1/2 \not< 1/2$
Bugeaud09.problem_10_9.variants.measure_zero Mahler [Mah68] proved that the set of Z-numbers has Lebesgue measure zero
Bugeaud09.problem_10_9.variants.dimH_lt_one Flatto, Lagarias, and Pollington [FLP95] proved that the set of Z-numbers has Hausdorff dimension strictly less than $1$
Equidistribution.IsAccumulationPoint A point x is an accumulation point of a sequence s_0, s_1, ... if any neighbourhood of x contains a point of the sequence distinct from x
Equidistribution.isAccumulationPoint_exists_subsequence_tendsto If a point x is an accumulation point of a sequence s_0, s_1, ... then there is a subsequence of s that tends to x
Equidistribution.isEquidistributedModuloOne_three_halves_pow The sequence (3/2)^n is equidistributed modulo 1
Equidistribution.isEquidistributedModuloOne_transcendental_three_halves_pow It is not true that for every transcendental number x the sequence x * (3 / 2) ^ n is equidistributed modulo 1
Equidistribution.isAccumulationPoint_three_halves_pow_infinite The sequence (3/2)^n has infinitely many accumulation points modulo 1
Equidistribution.isAccumulationPoint_three_halves_pow Find an accumulation point of the sequence (3/2)^n modulo 1
Equidistribution.fract_three_halves_pow_injective The values of (3/2)^n modulo 1 are pairwise distinct: if (3/2)^n - (3/2)^m were an integer for m < n, then 3^n - 3^m * 2^(n - m) would be even
Equidistribution.isAccumulationPoint_three_halves_pow_exists There is an accumulation point of the sequence (3/2)^n modulo 1: its values are pairwise distinct and lie in [0, 1], so they have an accumulation point by compactness
Erdos1.IsSumDistinctSet Sum-distinct sets
Erdos1.erdos_1 Erdos Problem 1: sum-distinct sets need N of order 2^n after Erdos1.IsSumDistinctSet
Erdos1.erdos_1.variants.weaker The trivial lower bound is $N \gg 2^n / n$
Erdos1.erdos_1.variants.lb Erdős and Moser [Er56] proved $$ N \geq (\tfrac{1}{4} - o(1)) \frac{2^n}{\sqrt{n}}. $$ [Er56] Erdős, P., Problems and results in additive number theory
Erdos1.IsSumDistinctRealSet A finite set of real numbers is said to be sum-distinct if all the subset sums differ by at least $1$
Erdos1.erdos_1.variants.real A generalisation of the problem to sets $A \subseteq (0, N]$ of real numbers, such that the subset sums all differ by at least $1$ is proposed in [Er73] and [ErGr80]
Erdos1.erdos_1.variants.least_N_3 The minimal value of $N$ such that there exists a sum-distinct set with three elements is $4$
Erdos1.erdos_1.variants.least_N_4 The minimal value of $N$ such that there exists a sum-distinct set with four elements is $7$
Erdos1.erdos_1.variants.least_N_5 The minimal value of $N$ such that there exists a sum-distinct set with five elements is $13$
Erdos1.erdos_1.variants.least_N_9 The minimal value of $N$ such that there exists a sum-distinct set with nine elements is $161$
Erdos10.sumPrimeAndTwoPows The set of natural numbers that can be written as a sum of a prime and at most $k$ powers of $2$
Erdos10.two_mem_sumPrimeAndTwoPows_zero A prime is the sum of a prime and no powers of $2$
Erdos10.one_not_mem_sumPrimeAndTwoPows $1$ is smaller than every prime, so it lies in no sumPrimeAndTwoPows k
Erdos10.erdos_10 Is there some $k$ such that every large integer is the sum of a prime and at most $k$ powers of $2$?
Erdos10.erdos_10.variants.gallagher Documented declaration: gallagher
Erdos10.erdos_10.variants.granville_soundararajan_odd Granville and Soundararajan [GrSo98] have conjectured that at most
Erdos10.erdos_10.variants.grechuk_example Bogdan Grechuk has observed that 1117175146 is not the sum of a prime and at most $3$ powers of $2$
Erdos10.erdos_10.variants.two_pows There are infinitely many even integers not the sum of a prime and $2$ powers of $2$
Erdos100.DistancesSeparated If two distances in A differ, they differ by at least 1
Erdos100.erdos_100 Is the diameter of $A$ at least $Cn$ for some constant $C > 0$?
Erdos100.erdos_100.variants.strong Stronger conjecture: diameter $\geq n - 1$ for sufficiently large $n$
Erdos100.erdos_100.variants.kanold From [Kanold]: diameter $\geq n^{3/4}$
Erdos100.erdos_100.variants.guth_katz From [GuKa15]: diameter $\gg n / \log n$
Erdos100.erdos_100_piepmeyer From [Piepmeyer]: 9 points with diameter $< 5$
Erdos1000.phiSeq Given an infinite sequence of positive integers
Erdos1000.phiAvg The average $\frac{1}{N}\sum_{k\leq N}\frac{\phi_A(k)}{n_k}$
Erdos1000.erdos_1000 Documented declaration: erdos_1000
Erdos1000.erdos_1000.variants.totient_le It is trivial that $\phi_A(k)\geq \phi(n_k)$, where $\phi$ is the Euler totient function
Erdos1000.erdos_1000.variants.liminf_eq_zero Documented declaration: liminf_eq_zero
Erdos1000.erdos_1000.variants.not_tendsto_zero Erdős [Er64b] proved that the limit of $\frac{\phi_A(k)}{n_k}$ as $k\to \infty$ cannot be $0$
Erdos1000.erdos_1000.variants.limsup_eq_one In fact he proved that if $\liminf \frac{\phi_A(k)}{n_k}=0$ then $\limsup \frac{\phi_A(k)}{n_k}=1$
Erdos1002.erdos_1002 For any $0<\alpha<1$, let $f(\alpha,n)=\frac{1}{\log n}\sum_{1\leq k\leq n}(\tfrac{1}{2}- \{ \alpha k\})$
Erdos1002.erdos_1002.variants.kesten Documented declaration: kesten
Erdos1003.erdos_1003 Are there infinitely many solutions to $\phi(n) = \phi(n+1)$, where $\phi$ is the Euler totient function?
Erdos1003.erdos_1003.variants.Icc Erdős [Er85e] says that, presumably, for every $k \geq 1$ the equation $$\phi(n) = \phi(n+1) = \cdots = \phi (n+k)$$ has infinitely many solutions
Erdos1003.erdos_1003.variants.eps87 Erdős, Pomerance, and Sárközy [EPS87] proved that for all large $x$, the number of $n \leq x$ with $\phi(n) = \phi(n+1)$ is at most $$\frac{x}{\exp((\log x)^{1/3})}$$
Erdos1004.IsDistinctTotientRun IsDistinctTotientRun n K means that the values φ(n+1), φ(n+2), ..., φ(n+K) are all distinct
Erdos1004.erdos_1004 For any fixed c > 0, if x is sufficiently large then there exists n ≤ x such that the values of φ(n+k) are all distinct for 1 ≤ k ≤ (log x)^c
Erdos1004.erdos_1004.variants.le_of_isDistinctTotientRun Erdős, Pomerance, and Sárközy [EPS87] proved that if φ(n+k) are all distinct for 1 ≤ k ≤ K then K ≤ n / exp(c (log n)^{1/3}) for some constant c > 0
Erdos1005.IsFarey A rational q is a Farey fraction of order n if it lies in [0, 1] and has denominator at most n
Erdos1005.betweenCount The number of Farey fractions of order n strictly between x and y
Erdos1005.f Documented declaration: f
Erdos1005.erdos_1005 Let $\frac{a_1}{b_1}, \frac{a_2}{b_2}, \ldots$ be the Farey fractions of order $n \geq 4$
Erdos1007.K133 The complete tripartite graph $K_{1,3,3}$
Erdos1007.erdos_1007 The dimension of a graph $G$ is the minimal $n$ such that $G$ can be embedded in $\mathbb{R}^n$ such that every edge of $G$ is a unit line segment
Erdos1008.erdos_1008 Does every graph with $m$ edges contain a subgraph with $\gg m^{2/3}$ edges which contains no $C_4$?
Erdos1008.erdos_1008.variants.three_quarters Originally asked by Bollobás and Erdős in 'a colloquium on graph theory at Tihany' with $m^{2/3}$ replaced by $m^{3/4}$
Erdos1008.erdos_1008.variants.folkman Folkman's counterexample $K_{n,n^2}$, which has $n^3$ edges, and yet every subgraph with $>n^2+\binom{n}{2}$ edges contains a $C_4$
Erdos1008.erdos_1008.variants.lower_bound In [Er71] Erdős revises the conjecture to $m^{2/3}$, and notes $\gg m^{1/2}$ is trivial
Erdos1009.erdos_1009.variants.sauer Documented declaration: sauer
Erdos101.linesWithPointsFor The set of lines in $\mathbb{R}^2$ containing exactly $k$ points from a given set $S$
Erdos101.numLinesWithFourPointMax The maximum number of lines containing exactly $4$ points among all sets $S$ of $n$ points in $\mathbb{R}^2$ satisfying the condition that no five points are collinear
Erdos101.erdos_101 Given $n$ points in $\mathbb{R}^2$, no five of which are on a line, the number of lines containing four points is $o(n^2)$
Erdos1010.erdos_1010.variants.rademacher Rademacher proved that every graph on $n$ vertices with $\lfloor n^2/4\rfloor+1$ edges contains at least $\lfloor n/2\rfloor$ triangles
Erdos1010.erdos_1010.variants.erdos Erdős [Er62d] proved that every graph on $n$ vertices with $\lfloor n^2/4\rfloor+t$ edges contains at least $t\lfloor n/2\rfloor$ triangles, for all $t0$
Erdos1014.erdos_1014 Let $R(k,l)$ be the Ramsey number, so the minimal $n$ such that every graph on at least $n$ vertices contains either a $K_k$ or an independent set on $l$ vertices
Erdos1014.erdos_1014.variants.upper_bound That proof in fact shows that $$R(k,l+1)\leq (1+O(l^{-c/k^2}))R(k,l)$$ for some constant $c>0$
Erdos102.pointsOn The number of points of P that lie on L
Erdos102.richLineCount The number of lines that contain more than three points of P
Erdos102.Admissible P has $n$ points and at least $c n^2$ lines that each contain more than three points of P
Erdos102.maxCollinear The largest number of points of P on a single line
Erdos102.h $h_c(n)$: the minimum of maxCollinear P over all admissible P
Erdos102.exists_line The plane contains a line
Erdos102.le_maxCollinear_iff M ≤ maxCollinear P exactly when some line contains at least M points of P
Erdos102.le_h_iff M ≤ h c n exactly when every admissible configuration of $n$ points has a line that contains at least M of its points
Erdos102.tendsto_h_iff The statement $h_c(n) \to \infty$, written without $h_c$
Erdos102.erdos_102 Documented declaration: erdos_102
Erdos102.erdos_102.variants.five It is not known whether $h_c(n) \geq 5$ for all sufficiently large $n$
Erdos102.erdos_102.variants.upper_sqrt Documented declaration: upper_sqrt
Erdos102.erdos_102.variants.hunter Erdős [Er95] suggested that perhaps $h_c(n) \gg_c n^{1/2}$
Erdos1020.f The maximum number of edges in an r-uniform hypergraph on n vertices containing no matching of size k (i.e. no k pairwise vertex-disjoint edges)
Erdos1020.erdos_1020 Let $f(n;r,k)$ be the maximal number of edges in an $r$-uniform hypergraph which contains no set of $k$ many independent edges
Erdos1021.cliqueSubdivision Documented declaration: cliqueSubdivision
Erdos1022.SparseImpliesPropertyB Documented declaration: SparseImpliesPropertyB
Erdos1022.erdos_1022.variants.lt_two Documented declaration: lt_two
Erdos1022.erdos_1022.variants.lovasz Erdős originally conjectured, in this language, that $c_2=1$, which he reports in [Er71] was proved by Lovász
Erdos1023.F F n is the maximal size of a family of subsets of $\{1,\ldots,n\}$ such that no set in this family is the union of other members of the family
Erdos1023.erdos_1023 Let $F(n)$ be the maximal size of a family of subsets of $\{1,\ldots,n\}$ such that no set in this family is the union of other members of the family
Erdos1023.erdos_1023.variants.erdos_kleitman Erdős and Kleitman proved in unpublished work that $$F(n)\asymp \frac{2^n}{n^{1/2}}.$$ ([Er71] has an exponent of $3/2$, but this is presumably a typo.)
Erdos1023.erdos_1023.variants.hunter Hunter observes in the comments that this follows from the solution to [447], which implies $F(n)\sim \binom{n}{n/2}$
Erdos1024.erdos_1024 Let $f(n)$ be such that every $3$-uniform linear hypergraph on $n$ vertices contains an independent set on $f(n)$ vertices
Erdos1025.IsIndependent X is independent for a set mapping f on pairs if f s(x, y) ∉ X whenever x ≠ y are in X
Erdos1025.g g n is the largest k such that every set mapping f on the pairs of an n-element set with f(x, y) ∉ {x, y} admits an independent set of size at least k
Erdos1026.monotonicSubsequenceSums The set of sums $\sum x_{i_r}$, where $x_{i_1},\ldots,x_{i_r}$ ranges over the monotonic subsequences of the sequence $x_1,\ldots,x_n$
Erdos1026.admissibleConstants The set of constants
Erdos1026.erdos_1026 Let $x_1,\ldots,x_n$ be a sequence of distinct real numbers
Erdos1026.erdos_1026.variants.upper_bound A construction of Cambie in the comments shows that $c\leq 1$
Erdos1026.erdos_1026.variants.lower_bound Hanani [Ha57] showed that every sequence is the disjoint union of at most $(\sqrt{2}+o(1))\sqrt{n}$ many monotonic subsequences, whence $c\geq 1/\sqrt{2}$
Erdos1026.erdos_1026.variants.hanani Hanani [Ha57] showed that every sequence is the disjoint union of at most $(\sqrt{2}+o(1))\sqrt{n}$ many monotonic subsequences
Erdos1026.erdos_1026.variants.weighted_erdos_szekeres Documented declaration: weighted_erdos_szekeres
Erdos1027.groundSet The ground set $X=\bigcup_{A\in\mathcal{F}}A$ of a finite family of finite sets
Erdos1027.goodSets The sets $B\subseteq X$ which intersect every set in $\mathcal{F}$, yet contain none of them
Erdos1028.imbalance The imbalance of a finite set X under a colouring f, that is, $\left\lvert \sum_{x<y\in X} f(x,y)\right\rvert$
Erdos1028.H H n is the minimum, over all colourings f of pairs from {1, …, n} with values in {-1, 1}, of the largest imbalance of a subset X ⊆ {1, …, n}
Erdos1028.erdos_1028 Let $$H(n)=\min_f \max_{X\subseteq \{1,\ldots,n\}} \left\lvert \sum_{x<y\in X} f(x,y)\right\rvert,$$ where $f$ ranges over all functions $f:\{1,\ldots,n\}^2\to \{-1,1\}$
Erdos1028.erdos_1028.variants.lower_bound Erdős [Er63d] proved $$\frac{n}{4}\leq H(n) \ll n^{3/2}.$$
Erdos1028.erdos_1028.variants.upper_bound Erdős [Er63d] proved $$\frac{n}{4}\leq H(n) \ll n^{3/2}.$$
Erdos1028.erdos_1028.variants.erdos_spencer Erdős and Spencer [ErSp71] proved that $H(n)\gg n^{3/2}$
Erdos1029.erdos_1029 Documented declaration: erdos_1029
Erdos103.separatedSets The sets of $n$ points in $\mathbb{R}^2$ whose pairwise distances are all at least $1$
Erdos103.IsMinimiser A is a minimiser for n if it is a separated set of n points whose diameter is at most the diameter of every separated set of n points (compare Erdos99.erdos_99)
Erdos103.Congruent Two finite sets of points are congruent if an isometry of the plane maps one onto the other
Erdos103.h Documented declaration: h
Erdos103.erdos_103 Let $h(n)$ count the number of incongruent sets of $n$ points in $\mathbb{R}^2$ which minimise the diameter subject to the constraint that $d(x,y)\geq 1$ for all points $x\neq y$
Erdos103.erdos_103.variants.two_le It is not even known whether $h(n)\geq 2$ for all large $n$
Erdos1030.erdos_1030 Let $R(k,l)$ be the usual Ramsey number: the smallest $n$ such that if the edges of $K_n$ are coloured red and blue then there exists either a red $K_k$ or a blue $K_l$
Erdos1034.JoinedToTwo JoinedToTwo G T Y holds when every vertex of Y is joined to at least two (distinct) vertices of T
Erdos1034.erdos_1034 Let $G$ be a graph on $n$ vertices with $>n^2/4$ many edges
Erdos1034.erdos_1034.variants.lower_bound Erdős and Faudree asked about the threshold
Erdos1034.erdos_1034.variants.upper_bound Documented declaration: upper_bound
Erdos1034.erdos_1034.variants.k4_free Erdős suggested that the answer is different if $G$ has no $K_4$
Erdos1035.erdos_1035 Is there a constant $c > 0$ such that every graph on $2^n$ vertices with minimum degree $> (1-c) \cdot 2^n$ contains the $n$-dimensional hypercube $Q_n$?
Erdos1036.HasManyNonIsomorphicInducedSubgraphs Documented declaration: HasManyNonIsomorphicInducedSubgraphs
Erdos1036.NoLargeBiclique Neither G nor its complement contains a balanced complete bipartite graph $K_{k,k}$ on more than m vertices as a (not necessarily induced) subgraph
Erdos1036.erdos_1036 Let $G$ be a graph on $n$ vertices which does not contain a trivial (empty or complete) graph on more than $c\log n$ vertices
Erdos1036.erdos_1036.variants.alon_hajnal Alon and Hajnal [AlHa91] proved that $G$ must contain at least $$\exp\left(n(\log n)^{-O(\log\log n)}\right)$$ many non-isomorphic induced subgraphs
Erdos1036.erdos_1036.variants.erdos_hajnal Erdős and Hajnal [ErHa89b] proved that if neither
Erdos1037.IsTrivialSet A set s of vertices of a graph G is trivial if the subgraph it induces is empty or complete, that is, if s is an independent set or a clique of G
Erdos1037.erdos_1037 Let $G$ be a graph on $n$ vertices in which every degree occurs at most twice, and the number of distinct degrees is $>(\frac{1}{2}+\epsilon)n$
Erdos1037.erdos_1037.variants.cambie_chan_hunter Documented declaration: cambie_chan_hunter
Erdos1038.erdos_1038.parts.i What is the infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1]?
Erdos1038.erdos_1038.parts.ii The supremum of |{x ∈ ℝ : |f x| < 1}| over all monic polynomials f such that all of its roots are real and contained in [-1,1] is 2 * 2 ^ (1 / 2)
Erdos1038.erdos_1038.variants.inf_upperBound The infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1] is < 1.835
Erdos1038.erdos_1038.varaints.inf_lowerBound The infimum of |{x ∈ ℝ : |f x| < 1}| over all nonconstant monic polynomials f such that all of its roots are real and contained in [-1,1] is ≥ 2 ^ (4 / 3) - 1
Erdos104.unitCircleCount The number of distinct unit circles containing at least three points of P
Erdos104.possibleUnitCircleCounts The set of unit-circle counts attained by configurations of n points in the plane
Erdos104.maxUnitCircleCount The maximum number of qualifying unit circles attained by a configuration of n points
Erdos104.erdos_104 Given $n$ points in $\mathbb{R}^2$ the number of distinct unit circles containing at least three points is $o(n^2)$
Erdos1041.length The length of a subset $s$ of $\mathbb{C}$ is defined to be its 1-dimensional Hausdorff measure $\mathcal{H}^1(s)$
Erdos1041.exists_connected_component_contains_two_roots Documented declaration: exists_connected_component_contains_two_roots
Erdos1043.levelSet The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert\leq 1\}$
Erdos1043.erdos_1043.variants.weak On the other hand, Pommerenke also proved there always exists a line such that the projection has measure at most 3.3
Erdos1044.IsAdmissible The polynomials under consideration: those of the form $f(z)=\prod_{i=1}^n(z-z_i)$, for some $n\geq 1$, where $\lvert z_i\rvert\leq 1$ for all $i$
Erdos1044.maxBoundaryLength Documented declaration: maxBoundaryLength
Erdos1044.erdos_1044 Let $f(z)=\prod_{i=1}^n(z-z_i)\in\mathbb{C}[x]$ where $\lvert z_i\rvert\leq 1$ for all $i$
Erdos1044.erdos_1044.variants.infimum_eq_two This has been resolved by Tang, who proved that the infimum of $\Lambda(f)$ over all such $f$ is $2$
Erdos1044.erdos_1044.variants.fixed_degree Tang also suggests that, if the degree $n$ is fixed, then the infimum over all such $f$ of degree $n$ is attained by $f_n(z)=z^n-1$
Erdos1044.erdos_1044.variants.fixed_degree_of_le_two Tang also suggests that, if the degree $n$ is fixed, then the infimum over all such $f$ of degree $n$ is attained by $f_n(z)=z^n-1$ (and proves this for $n=1$ and $n=2$)
Erdos1046.lemniscate The open unit lemniscate $E = \{z : \lvert f(z)\rvert < 1\}$ of a complex polynomial
Erdos1046.closedLemniscate The closed unit lemniscate $\{z : \lvert f(z)\rvert \leq 1\}$ of a complex polynomial
Erdos1046.erdos_1046.variants.diameter Erdős, Herzog, and Piranian [EHP58] conjecture that if $\{ z: \lvert f(z)\rvert\leq 1\}$ is connected then its diameter is at least $2$
Erdos1047.sublevelSet The sublevel set $\{ z: \lvert f(z)\rvert\leq c\}$ of a polynomial $f\in\mathbb{C}[x]$
Erdos1047.strictSublevelSet The strict sublevel set $\{ z: \lvert f(z)\rvert< c\}$ of a polynomial $f\in\mathbb{C}[x]$
Erdos1047.componentsIn The connected components of a set s ⊆ ℂ, viewed as subsets of ℂ
Erdos1047.erdos_1047.variants.pommerenke Documented declaration: pommerenke
Erdos1047.erdos_1047.variants.goodman Goodman [Go66] proved that one of the three components of $$\{ z: \lvert (z^2+1)(z-2)^2\rvert < 5^{3/2}/4\}$$ is not convex
Erdos1047.erdos_1047.variants.goodman_simple_roots Goodman [Go66] constructed an example with simple roots, of degree $4$
Erdos1047.erdos_1047.variants.referee The referee of the paper [Go66] also gave the example of $\lvert z(z^5-1)\rvert< 5\cdot 6^{-6/5}$
Erdos1048.openLevelSet The set $\{ z \in \mathbb{C} : \lvert f(z)\rvert < 1\}$
Erdos1048.erdos_1048 Documented declaration: erdos_1048
Erdos1048.erdos_1048.variants.pommerenke_ncard_components Documented declaration: pommerenke_ncard_components
Erdos1048.erdos_1048.variants.pommerenke_diam_tendsto_zero Documented declaration: pommerenke_diam_tendsto_zero
Erdos1048.erdos_1048.variants.r_le_one On the other hand, if $0<r\leq 1$, then the answer is yes, as also shown by Pommerenke [Po61]
Erdos1048.erdos_1048.variants.diam_ge_two If $0\leq r\leq 1/2$ then the component which contains $0$ must have diameter $\geq 2$, which $f(z)=z^n$ shows is best possible
Erdos1048.erdos_1048.variants.diam_ge_two_is_best If $0\leq r\leq 1/2$ then the component which contains $0$ must have diameter $\geq 2$, which $f(z)=z^n$ shows is best possible
Erdos1048.erdos_1048.variants.diam_gt_inv_r If $1/21/r$
Erdos1048.erdos_1048.variants.diam_gt_two_sub_sq If $\frac{\sqrt{5}-1}{2}\leq r\leq 1$ then the component which contains $0$ must have diameter $>2-r^2$
Erdos1049.erdos_1049 Let $t>1$ be a rational number
Erdos1049.lambert_convergent Convergent case (|t| > 1)
Erdos1049.lambert_divergent Divergent case (|t| ≤ 1)
Erdos1049.lambert_series_eq_num_divisor_sum The classical Lambert series identity: $\sum_{n=1}^\infty \frac{1}{t^n - 1} = \sum_{n=1}^\infty \frac{\tau(n)}{t^n}$, where $\tau(n)$ counts the divisors of $n$
Erdos105.erdos_105 Let $A,B\subset \mathbb{R}^2$ be disjoint sets of size $n$ and $n-3$ respectively, with not all of $A$ contained on a single line
Erdos105.erdos_105.variants.hickerson A construction of Hickerson shows that this fails with $n-2$
Erdos105.erdos_105.variants.beck_szemeredi_trotter A result independently proved by Beck [Be83] and Szemerédi and Trotter [SzTr83] (see [211]) implies it is true with $n-3$ replaced by $cn$ for some constant $c>0$
Erdos105.erdos_105.variants.sub_four It remains possible that this holds with $n-4$ (or in general with $n-O(1)$ or $(1-o(1))n$)
Erdos1050.erdos_1050.variants.transcendental Erdős Problem 1050, variant [Er88c] (open)
Erdos1051.GrowthCondition A sequence of integers a satisfies the growth condition if $\liminf a_n^{\frac{1}{2^n}} > 1$
Erdos1051.ErdosSeries The series $\sum_{n=0}^\infty \frac{1}{a_n \cdot a_{n+1}}$
Erdos1051.erdos_1051 Documented declaration: erdos_1051
Erdos1051.erdos_1051.variants.rapid_growth Erdős [Er88c] notes that if the sequence grows rapidly to infinity (specifically, if $a_{n+1} \geq C \cdot a_n^2$ for some constant $C > 0$), then the series is irrational
Erdos1052.properUnitaryDivisors A proper unitary divisor of $n$ is a divisor $d$ of $n$ such that $d$ is coprime to $n/d$, and $d < n$
Erdos1052.IsUnitaryPerfect A number $n > 0$ is a unitary perfect number if it is the sum of its proper unitary divisors
Erdos1052.erdos_1052 Are there only finitely many unitary perfect numbers?
Erdos1052.even_of_isUnitaryPerfect All unitary perfect numbers are even
Erdos1054.f Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$
Erdos1054.erdos_1054.parts.i Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$
Erdos1054.erdos_1054.parts.ii Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$
Erdos1054.erdos_1054.parts.iii Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$
Erdos1054.f_undefined_at_2 Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$
Erdos1054.f_undefined_at_3 Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$
Erdos1055.IsOfClass A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$
Erdos1055.exists_p A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$
Erdos1055.p A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$
Erdos1055.erdos_1055 A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$
Erdos1055.erdos_1055.variants.erdos_limit A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$
Erdos1055.erdos_1055.variants.selfridge_limit A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$
Erdos1056.AllModProdEqualsOne The proposition that the modular product of a collection of consecutive interval equals
Erdos1056.erdos_1056 Let $k ≥ 2$
Erdos1056.erdos_1056.variants.k2 This is problem A15 in Guy's collection [Gu04], where he reports that in a letter in 1979 Erdős observed that $3 * 4 \equiv 5 * 6 * 7 \equiv 1 \mod 11$
Erdos1056.erdos_1056.variants.k3 Makowski [Ma83] found, for $k=3$: $2 * 3 * 4 * 5 \equiv 6 * 7 * 8 * 9 * 10 * 11 \equiv 12 * 13 * 14 * 15 \equiv 1 \mod 17$
Erdos1056.erdos_1056.variants.noll_simmons Noll and Simmons asked, more generally, whether there are solutions to $q_1! \equiv \dots \equiv q_k! \mod p$ for arbitrarily large $k$ (with $q_1 < \dots < q_k$)
Erdos1057.carmichaelCounting Let $C(x)$ count the number of Carmichael numbers in the interval $[1,x]$
Erdos1057.erdos_1057 Is it true that $C(x)=x^{1-o(1)}$?
Erdos1057.erdos_1057.variants.upper_bound Erdős [Er56c] proved $C(x) < x \exp\left(-c \frac{\log x\log\log\log x}{\log\log x}\right)$ for some constant $c>0$
Erdos1057.erdos_1057.variants.pomerance Pomerance [Po89] gave a heuristic suggesting that this is the true order of growth, and in fact $C(x)= x \exp\left(-(1+o(1))\frac{\log x\log\log\log x}{\log\log x}\right)$
Erdos1057.erdos_1057.variants.agp_infinite Alford, Granville, and Pomerance [AGP94] proved that $C(x)\to \infty$
Erdos1057.erdos_1057.variants.agp_lower_bound Alford, Granville, and Pomerance [AGP94] proved that $C(x)>x^{2/7}$ for large $x$
Erdos1057.erdos_1057.variants.harman_lower_bound The lower bound $C(x)> x^{0.33336704}$ was proved by Harman [Ha08]
Erdos1057.erdos_1057.variants.lichtman_lower_bound This exponent was improved to $0.3389$ by Lichtman [Li22]
Erdos1058.IsSolution n is a solution if n ∈ [p_{k-1}, p_k) for some k (with the convention p_0 = 1, where p_1 = 2 < p_2 < ⋯ are the primes) and the only primes dividing `n!
Erdos1059.erdos_1059 Are there infinitely many primes $p$ such that $p - k!$ is composite for each $k$ such that $1 ≤ k! < p$?
Erdos1060.erdos_1060.parts.i The conjecture is about the function $f(n)$ which counts the number of solutions to $k\sigma(k)=n$, where $\sigma(k)$ is the sum of divisors of $k$
Erdos1060.erdos_1060.parts.ii Part (ii) of Erdős Problem 1060: bound on the number of $k \le n$ with $k \sigma_1(k) = n$
Erdos1061.S Let S x count the number of ordered pairs of positive integers (a, b) with a + b ≤ x such that σ(a) + σ(b) = σ(a + b), where σ is the sum of divisors function
Erdos1061.erdos_1061 How many (ordered) solutions are there to σ(a) + σ(b) = σ(a + b) with a + b ≤ x?
Erdos1062.ForkFree A set A of positive integers is fork-free if no element divides two distinct other elements of A
Erdos1062.f The extremal function from Erdős problem 1062: the largest size of a fork-free subset of {1,...,n}
Erdos1062.erdos_1062.parts.ii Erdős asked whether the limiting density f n / n exists and, if so, whether it is irrational
Erdos1062.erdos_1062.variants.lower_bound The interval [⌊n/3⌋, n] is fork-free, and therefore f n is at least ⌈2n / 3⌉
Erdos1062.erdos_1062.variants.lebensold_bounds Lebensold proved that for large n, the function f n lies between 0.6725 n and 0.6736 n
Erdos1063.n Let $n_k$ be the least $n \ge 2k$ such that all but one of the integers $n - i$ with $0 \le i < k$ divide $\binom{n}{k}$
Erdos1063.erdos_1063.better_upper Estimate $n_k$ by finding a better upper bound than Cambie's $n_k \leq k \cdot \operatorname{lcm}(1, \dotsc, k-1)$
Erdos1063.erdos_1063.variants.small_values The initial values satisfy $n_2 = 4$, $n_3 = 6$, $n_4 = 9$, and $n_5 = 12$ ([Gu04], Problem B31)
Erdos1064.erdos_1064 Let $ϕ(n)$ be the Euler's totient function, then the $n$ satisfies $ϕ(n)>ϕ(n - ϕ(n))$ have asymptotic density 1
Erdos1064.erdos_1064.variants.k2 Let $ϕ(n)$ be the Euler's totient function, there exist infinitely many $n$ such that $ϕ(n)< ϕ(n - ϕ(n))$ Reference: [GLW01] Grytczuk, A
Erdos1064.erdos_1064.variants.general_function For any function $f(n)=o(n)$, we have $\phi(n)>\phi(n-\phi(n))+f(n)$ for almost all $n$
Erdos1065.erdos_1065.parts.i Are there infinitely many primes $p$ such that $p = 2^k * q + 1$ for some prime $q$ and $k ≥ 0$?
Erdos1065.erdos_1065.parts.ii Are there infinitely many primes $p$ such that $p = 2^k 3^l q + 1$ for some prime $q$ and $k ≥ 0$, $l ≥ 0$?
Erdos1067.InfinitelyEdgeConnected A graph is infinitely edge-connected if to disconnect the graph requires deleting infinitely many edges
Erdos1067.erdos_1067 Does every graph with chromatic number $\aleph_1$ contain an infinitely connected subgraph with chromatic number $\aleph_1$?
Erdos1067.erdos_1067.variants.infinite_edge_connectivity Does every graph with chromatic number $\aleph_1$ contain an infinitely edge-connected subgraph with chromatic number $\aleph_1$?
Erdos1068.erdos_1068 Does every graph with chromatic number $\aleph_1$ contain a countable subgraph which is infinitely connected?
Erdos107.cardSet The set of $N$ such that any $N$ points in the plane, no three on a line, contain a convex $n$-gon
Erdos107.f The function $f(n)$ specified in erdos_107
Erdos107.erdos_107 Let $f(n)$ be minimal such that any $f(n)$ points in $ℝ^2$, no three on a line, contain $n$ points which form the vertices of a convex $n$-gon
Erdos107.nonempty_cardSet For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line, contain a convex $n$-gon
Erdos107.f_zero_eq Depending on details of definitions, the statement is false or trivial for $n < 3$
Erdos107.convexIndep_triple_of_not_collinear Three distinct non-collinear points form a convex-independent set
Erdos107.variants.ersz_bounds Erdős and Szekeres proved the bounds $$ 2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1 $$ ([ErSz60] and [ErSz35] respectively)
Erdos107.variants.su_bound Suk [Su17] proved $$ f(n) ≤ 2^{(1+o(1))n}. $$ [Su17] Suk, Andrew, On the Erdős-Szekeres convex polygon problem
Erdos107.variants.hmpt_bound The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove $$ f(n) ≤ 2^{n+O(\sqrt{n\log n})}. $$ [HMPT20] Holmsen, Andreas F
Erdos1071.SegmentsDisjoint Two segments are disjoint if they only intersect at their endpoints (if at all)
Erdos1071.erdos_1071.parts.i Can a finite set of disjoint unit segments in a unit square be maximal?
Erdos1071.erdos_1071.parts.ii Is there a region $R$ with a maximal set of disjoint unit line segments that is countably infinite?
Erdos1072.f For any prime $p$, let $f(p)$ be the least integer such that $f(p)! + 1 \equiv 0 \mod p$
Erdos1072.erdos_1072.parts.i Is it true that there are infinitely many $p$ for which $f(p) = p − 1$?
Erdos1072.erdos_1072.parts.ii Is it true that $f(p)/p \to 0$ for $p \to \infty$ in a density 1 subset of the primes?
Erdos1072.erdos_1072.variants.littleo Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \le x$ for which $f(p)=p−1$ is $o(x/\log x)$
Erdos1073.A Let $A(x)$ count the number of composite $u < x$ such that $n!+1 \equiv 0 (\mod u)$ for some $n$
Erdos1073.erdos_1073 Is it true that $A(x) \le x^{o(1)}$?
Erdos1074.EHSNumbers The EHS numbers (after Erdős, Hardy, and Subbarao) are those $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$
Erdos1074.PillaiPrimes The Pillai primes are those primes $p$ such that there exists an $m \ge 1$ with $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$
Erdos1074.erdos_1074.parts.i Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$
Erdos1074.erdos_1074.parts.ii Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$
Erdos1074.erdos_1074.parts.iii Documented declaration: iii
Erdos1074.erdos_1074.parts.iv Documented declaration: iv
Erdos1074.erdos_1074.variants.mem_pillaiPrimes Pillai [Pi30] raised the question of whether there exist any primes in $P$
Erdos1074.erdos_1074.variants.PillaiPrimes_infinite Erdős, Hardy, and Subbarao proved that $P$ is infinite
Erdos1074.erdos_1074.variants.EHSNumbers_init The sequence $S$ begins $8, 9, 13, 14, 15, 16, 17, ...$
Erdos1074.erdos_1074.variants.PillaiPrimes_init The sequence $P$ begins $23, 29, 59, 61, 67, 71, ...$
Erdos1074.erdos_1074.variants.EHSNumbers_one Documented declaration: EHSNumbers_one
Erdos1075.erdos_1075 Let $r\geq 3$
Erdos1079.erdos_1079 Let $r\geq 4$
Erdos108.erdos_108 For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r) contains a subgraph of girth ≥ r and chromatic number ≥ k?
Erdos1080.IsBipartition IsBipartition G X Y means that X and Y form a bipartition of the vertices of G
Erdos1080.erdos_1080 Let $G$ be a bipartite graph on $n$ vertices such that one part has $\lfloor n^{2/3}\rfloor$ vertices
Erdos1082.erdos_1082.parts.i Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line
Erdos1082.erdos_1082.parts.ii Let $A\subset \mathbb{R}^2$ be a set of $n$ points with no three on a line
Erdos1083.erdos_1083 Let $d\geq 3$, and let $f_d(n)$ be the minimal $m$ such that every set of $n$ points in $\mathbb{R}^d$ determines at least $m$ distinct distances
Erdos1084.f The maximal number of pairs of points which are distance 1 apart that a set of n 1-separated points in ℝ^d make
Erdos1084.erdos_1084.variants.upper_d1 It is easy to check that $f_1(n) = n - 1$
Erdos1084.erdos_1084.variants.easy_upper_d2 It is easy to check that $f_2(n) < 3n$
Erdos1084.erdos_1084.variants.upper_d2 Erdős showed that there is some constant $c > 0$ such that $f_2(n) < 3n - c n^{1/2}$
Erdos1084.erdos_1084.variants.triangular_optimal_d2 Erdős conjectured that the triangular lattice is best possible in 2D, in particular that $f_2(3n^2 + 3n + 1) = 9n^2 + 3n$
Erdos1084.erdos_1084.variants.upper_lower_d3 Erdős claims the existence of two constants $c_1, c_2 > 0$ such that $6n - c_1 n^{2/3} ≤ f_3(n) \le 6n - c_2 n^{2/3}$
Erdos1085.f The maximal number of pairs of points which are distance 1 apart that a set of n points in ℝ^d make
Erdos1085.erdos_1085.variants.lower_d2 Erdős showed $f_2(n) > n^{1+c/\log\log n}$ for some $c > 0$
Erdos1085.erdos_1085.variants.upper_d2 Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$
Erdos1085.erdos_1085.variants.lower_d3 Erdős showed $f_3(n) = Ω(n^{4/3}\log\log n)$
Erdos1085.erdos_1085.variants.upper_d3 Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound?
Erdos1085.erdos_1085.variants.lower_d4_lenz Lenz showed that, for $d \ge 4$, $f_d(n) \ge \frac{p - 1}{2p} n^2 - O(1)$ where $p = \lfloor\frac d2\rfloor$
Erdos1085.erdos_1085.variants.upper_d4_erdos Erdős showed that, for $d \ge 4$, $f_d(n) \le \left(\frac{p - 1}{2p} + o(1)\right) n^2$ where $p = \lfloor\frac d2\rfloor$
Erdos1085.erdos_1085.variants.upper_lower_d5_odd Documented declaration: upper_lower_d5_odd
Erdos1088.PairwiseDistancesDistinct A finite set of points has all pairwise distances distinct
Erdos1088.HasSubsetWithDistinctDistances The finite set S contains an n-point subset with all pairwise distances distinct
Erdos1088.cardSet The set of cardinalities m such that every m-point subset of ℝ^d contains an n-point subset with all pairwise distances distinct
Erdos1088.f The least m such that every m-point subset of ℝ^d contains an n-point subset with all pairwise distances distinct
Erdos1088.f_isLeast f d n is the least cardinality with the required distinct-distance property
Erdos1088.erdos_1088 Let $f_d(n)$ be the minimal $m$ such that any set of $m$ points in $\mathbb{R}^d$ contains a set of $n$ points for which any two determined distances are distinct
Erdos1089.g Documented declaration: g
Erdos1090.erdos_1090 Let $k\geq 3$
Erdos1092.IsAdmissible An edge-budget function
Erdos1092.f_asymptotic_2 Is it true that $f_2(n) \gg n$?
Erdos1092.f_asymptotic_general More generally, is $f_r(n)\gg_r n$ for every $r \geq 2$?
Erdos1093.deficiency If defined, the deficiency is the count of $0 \le i < k$ such that $n - i$ is $k$-smooth, that is, divisible only by primes $\le k$
Erdos1093.erdos_1093.parts.i Are there infinitely many binomial coefficients with deficiency 1?
Erdos1093.erdos_1093.parts.ii Are there only finitely many binomial coefficients with deficiency > 1?
Erdos1094.erdos_1094 For all $n\ge 2k$ the least prime factor of $\binom{n}{k}$ is $\le\max(n/k,k)$, with only finitely many exceptions
Erdos1095.g Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$
Erdos1095.erdos_1095.variants.lower_solved The current record is $g(k) \gg \exp(c(\log k)^2)$ for some $c>0$, due to Konyagin [Ko99b]
Erdos1095.erdos_1095.variants.upper_conjecture Ecklund, Erdős, and Selfridge [EES74] proved $g(k)\leq \exp((1+o(1))k)$
Erdos1095.erdos_1095.variants.lower_conjecture Erdős, Lacampagne, and Selfridge [ELS93] write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$
Erdos1095.erdos_1095.variants.log_equivalent Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$
Erdos1097.CommonDifferencesThreeTermAP Documented declaration: CommonDifferencesThreeTermAP
Erdos1097.erdos_1097.variants.weaker A weaker bound has been proven: there are always at most $n^2$ such values of $d$
Erdos1097.erdos_1097.variants.lower_bound A trivial lower bound: for sufficiently large n there exist sets
Erdos1098.nonCommutingGraph The non-commuting graph
Erdos1098.erdos_1098 Documented declaration: erdos_1098
Erdos1098.erdos_1098.variants.center_finiteIndex Neumann [Ne76] proved that $\Gamma$ contains no infinite complete subgraph if and only if the centre of the group has finite index
Erdos1098.erdos_1098.variants.upper_bound Neumann [Ne76] noted that if the centre has index $n$ then $\Gamma$ contains no complete subgraph on $>n$ vertices
Erdos1099.h For $\alpha>1$ and the divisors $1=d_1<\cdots<d_{\tau(n)}=n$ of $n$, $$h_\alpha(n) = \sum_i \left( \frac{d_{i+1}}{d_i}-1\right)^\alpha.$$
Erdos1099.erdos_1099.variants.factorial Is $h_\alpha(n!)$ bounded?
Erdos1099.erdos_1099.variants.lcm Is $h_\alpha(\mathrm{lcm}(1,\ldots,n))$ bounded?
Erdos1099.erdos_1099.variants.ratio_sum Erdős remarks that this problem occurred to him when considering $\sum_i \frac{d_{i+1}}{d_i}$
Erdos11.erdos_11 Is every large odd integer $n$ the sum of a squarefree number and a power of 2?
Erdos11.erdos_11.variants.all_odd Every odd $n > 1$ is the sum of a squarefree number and a power of 2
Erdos11.eventually_of_all_odd The form for every odd $n > 1$ implies the form for large odd $n$
Erdos11.erdos_11.variants.not_four_dvd Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4
Erdos11.erdos_11.variants.two_pow_two Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?
Erdos11.erdos_11.variants.finite_bound1 Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2
Erdos11.erdos_11.variants.finite_bound2 Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2
Erdos11.erdos_11.variants.granville_soundararajan Suppose that every odd $n > 1$ is the sum of a squarefree number and a power of 2
Erdos1101.ASet The set of integers not divisible by any u_i
Erdos1101.A The sequence of integers A_u which are not divisible by any u_i arranged in a monotonic sequence
Erdos1101.t t_x such that u_0
Erdos1101.IsGood A sequence is "good" if 1
Erdos1101.erdos_1101.parts.i 1
Erdos1101.erdos_1101.parts.ii 2
Erdos1102.HasPropertyP Property P : A set $A ⊆ ℕ $ has property P, if for all $n ≥ 1$ the set $ \{a ∈ A | n + a\text{ is squarefree}\}$ is finite
Erdos1102.HasPropertyQ Property Q : A set $A ⊆ ℕ $ has property Q, if the set $\{n ∈ ℕ | ∀ a ∈ A, n > a\text{ implies }n + a\text{ is squarefree}\}$ is infinite
Erdos1104.triangleFreeMaxChromatic Maximum chromatic number of a triangle-free graph on n vertices
Erdos1104.erdos_1104.variants.lower Lower bound (Hefty–Horn–King–Pfender 2025)
Erdos1104.erdos_1104.variants.upper Upper bound (Davies–Illingworth 2022)
Erdos1105.erdos_1105.parts.i The anti-Ramsey number
Erdos1105.erdos_1105.parts.ii Let $P_k$ be the path on $k$ vertices and $\ell=\lfloor\frac{k-1}{2}\rfloor$
Erdos1106.p The partition function p(n) is the number of ways to write n as a sum of positive integers (where the order of the summands does not matter)
Erdos1106.erdos_1106.parts.i Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity
Erdos1106.erdos_1106.parts.ii Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$
Erdos1107.SumOfRPowerful Helper Property: $n$ is the sum of at most $r+1$ numbers, each of which is $r$-full
Erdos1107.erdos_1107 Let $r \ge 2$
Erdos1107.erdos_1107.variants.two Heath-Brown [He88] proved every large integer the sum of at most three $2$-powerful numbers
Erdos1108.FactorialSums The set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}_{\geq 1}\text{ finite}\right\}$ of all finite sums of distinct factorials
Erdos1108.erdos_1108.parts.i Documented declaration: i
Erdos1108.erdos_1108.parts.ii Does the set
Erdos1109.f f N is the largest size of a subset A ⊆ {1, ..., N} such that every element of A + A is squarefree
Erdos1109.erdos_1109 Let $f(N)$ be the size of the largest subset $A\subseteq \{1,\ldots,N\}$ such that every $n\in A+A$ is squarefree
Erdos1109.erdos_1109.variants.polylog Is the stronger polylogarithmic bound $f(N) \leq (\log N)^{O(1)}$ true?
Erdos1109.erdos_1109.variants.erdos_sarkozy_lower Erdős and Sárközy [ErSa87] proved the lower bound $\log N \ll f(N)$
Erdos1109.erdos_1109.variants.erdos_sarkozy_upper Erdős and Sárközy [ErSa87] proved the upper bound $f(N) \ll N^{3/4}\log N$
Erdos1109.erdos_1109.variants.konyagin_lower Konyagin [Ko04] improved the lower bound to $\log\log N(\log N)^2 \ll f(N)$
Erdos1109.erdos_1109.variants.konyagin_upper Konyagin [Ko04] improved the upper bound to $f(N) \ll N^{11/15+o(1)}$
Erdos1110.Representable $n$ is representable with respect to $p$ and $q$ if it is the sum of a finite divisibility antichain of terms of the form $p^kq^l$
Erdos1110.erdos_1110 Let $p>q\geq 2$ be two coprime integers
Erdos1113.HasFinitePrimeCoveringSet A covering set for a positive odd integer $k$ is a finite set of primes $P$ such that every number of the form $k \cdot 2^n + 1$ is divisible by at least one prime in $P$
Erdos1113.erdos_1113 Erdős Problem 1113
Erdos1113.erdos_1113.variants.filaseta_finch_kozek Filaseta–Finch–Kozek conjecture (2008)
Erdos1119.erdos_1119 Let $\mathfrak{m}$ be an infinite cardinal with $\aleph_0 < \mathfrak{m} < \mathfrak{c} = 2^{\aleph_0}$
Erdos1119.erdos_1119.variants.erdos_wetzel Documented declaration: erdos_wetzel
Erdos1119.erdos_1119.variants.erdos_wetzel_ch Documented declaration: erdos_wetzel_ch
Erdos1119.erdos_1119.test.empty_family Sanity check: the empty family of entire functions satisfies both the value-bound hypothesis and the cardinality conclusion of erdos_1119, for any infinite $\mathfrak{m}$
Erdos1121.erdos_1121 Documented declaration: erdos_1121
Erdos1121.erdos_1121.variants.higher_dimension Documented declaration: higher_dimension
Erdos1125.erdos_1125 Let $f:\mathbb{R}\to \mathbb{R}$ be such that $$2f(x) \leq f(x+h)+f(x+2h)$$ for every $x\in \mathbb{R}$ and $h>0$
Erdos1126.erdos_1126 Documented declaration: erdos_1126
Erdos1128.IsMonochromaticBox Documented declaration: IsMonochromaticBox
Erdos1128.Omega1 The set of countable ordinals, expressed using Mathlib's ω_ 1
Erdos1128.countable_Iio_of_lt_omega1 The set of ordinals strictly below any $\gamma < \omega_1$ is countable
Erdos1128.countable_Iio_omega1 Initial segments of ω₁ (as sets of Omega1 elements) are countable
Erdos1128.countable_subset_bdd Any countable subset of $\omega_1$ is bounded strictly below some element of $\omega_1$
Erdos1128.erdos_1128.prikryMills Prikry–Mills counterexample (key lemma): There exists a 2-colouring
Erdos1128.erdos_1128 Documented declaration: erdos_1128
Erdos1128.erdos_1128.variants.prikryMills_explicit Documented declaration: prikryMills_explicit
Erdos1128.erdos_1128.variants.two_dimensional_false The claim that every 2-colouring of $\omega_1 \times \omega_1$ has an uncountable monochromatic product rectangle is false in ZFC
Erdos1133.erdos_1133 Let $C>0$
Erdos1133.erdos_1133.variants.weaker Documented declaration: weaker
Erdos1134.A The smallest set of natural numbers which contains 1 and is closed under the operations x ↦ 2x + 1, x ↦ 3x + 1 and x ↦ 6x + 1
Erdos1135.erdos_1135 The Collatz conjecture states that for any positive integer $n$, there exists a natural number $m$ such that the $m$-th term of the sequence is 1
Erdos1136.AvoidsPowersOfTwo A set A of natural numbers has the property in the question if a + b ≠ 2 ^ k for all a, b ∈ A (not necessarily distinct) and all k ≥ 0
Erdos1136.muellerSet The set of all integers congruent to $3\cdot 2^i\pmod{2^{i+2}}$ for some $i\geq 0$
Erdos1136.erdos_1136 Does there exist $A\subset \mathbb{N}$ with lower density $>1/3$ such that $a+b\neq 2^k$ for any $a,b\in A$ and $k\geq 0$?
Erdos1136.erdos_1136.variants.multiples_of_three Achieving density $1/3$ is trivial, taking $A$ to be all multiples of $3$
Erdos1136.erdos_1136.variants.mueller Müller [Mu11] settled this question in the affirmative: in fact one can take
Erdos1136.erdos_1136.variants.upper_bound Müller also proved this is best possible, in that $A$ with the property in the question has lower density at most $1/2$
Erdos1137.erdos_1137 Let $d_n=p_{n+1}-p_n$, where $p_n$ denotes the $n$th prime
Erdos1138.sup_primeGap The maximal prime gap below $x$, i.e. $d(x) = \max_{p_n < x}(p_{n+1} - p_n)$, where $p_n$ denotes the $n$-th prime
Erdos1138.snd_gt_half_fst The filter on $\mathbb{R} \times \mathbb{R}$ corresponding to sending $x \to \infty$ subject to $x/2 < y < x$
Erdos1138.primeCount_Ioc_mul_const Given a pair $(x,y)$, this is the amount of primes in the interval above $y$, of length equalling the largest prime gap before $x$, scaled by a constant $C$
Erdos1139.erdos_1139 Let $1\leq u_1 < u_2 < \cdots$ be the sequence of integers with at most $2$ prime factors
Erdos114.lemniscate The lemniscate $\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}$ of a complex polynomial $p$
Erdos114.lemniscateLength The length of the lemniscate of $p$, as its one-dimensional Hausdorff measure
Erdos114.erdos_114 If $p(z) \in \mathbb{C}[z]$ is a monic polynomial of degree $n$, is the length of the curve $\{z \in \mathbb{C} : \lvert p(z)\rvert = 1\}$ maximised when $p(z) = z^n - 1$?
Erdos114.erdos_114.variants.large_n Tao [Ta25] proved that $z^n - 1$ maximises the length of the lemniscate for all sufficiently large $n$
Erdos114.erdos_114.variants.two Eremenko and Hayman [ErHa99] proved the case $n = 2$
Erdos114.erdos_114.variants.one The case $n = 1$
Erdos1141.residuePrimeUpperBound The cutoff $m^{1/4 + \varepsilon}$ of Theorem 1.3 of [Po17]
Erdos1141.residuePrimesUpTo The primes $\ell \leq m^{1/4+\varepsilon}$ with $\chi(\ell) = 1$
Erdos1141.erdos_1141.variants.pollack_1_3 Theorem 1.3 of [Po17]: for a quadratic character to a large enough modulus, the primes below
Erdos1141.erdos_1141.variants.mertens_third Mertens' third theorem [Me1874], in the weakened form the linked proof assumes: the product over the primes up to $n$ of $1 - 1/p$ is at least $1/(3\log n)$
Erdos1141.Erdos1141Prop The property that $n-k^2$ is prime for all $k$ with $(n,k)=1$ and $k^2 < n$
Erdos1142.Erdos1142Prop The property that $n > 2$ and $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$
Erdos1142.erdos_1142 Are there infinitely many $n > 2$ such that $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$?
Erdos1142.erdos_1142.variants.mientka_weitzenkamp Mientka and Weitzenkamp [MiWe69] proved that the only $n \leq 2^{44}$ such that $n > 2$ and $n - 2^k$ is prime for all $k \geq 1$ with $2^k < n$ are $4, 7, 15, 21, 45, 75, 105$
Erdos1142.erdos_1142.test_4 $4$ satisfies the Erdős 1142 property: $4 - 2 = 2$ is prime
Erdos1142.erdos_1142.test_7 $7$ satisfies the Erdős 1142 property: $7 - 2 = 5$ and $7 - 4 = 3$ are prime
Erdos1142.erdos_1142.test_15 $15$ satisfies the Erdős 1142 property: $15 - 2 = 13$, $15 - 4 = 11$, $15 - 8 = 7$
Erdos1142.erdos_1142.test_21 $21$ satisfies the Erdős 1142 property: $21 - 2 = 19$, $21 - 4 = 17$, $21 - 8 = 13$, $21 - 16 = 5$
Erdos1142.erdos_1142.test_45 $45$ satisfies the Erdős 1142 property: $45 - 2 = 43$, $45 - 4 = 41$, $45 - 8 = 37$, $45 - 16 = 29$, $45 - 32 = 13$
Erdos1142.erdos_1142.test_75 $75$ satisfies the Erdős 1142 property: $75 - 2 = 73$, $75 - 4 = 71$, $75 - 8 = 67$, $75 - 16 = 59$, $75 - 32 = 43$, $75 - 64 = 11$
Erdos1142.erdos_1142.test_105 $105$ satisfies the Erdős 1142 property: the largest known example
Erdos1142.erdos_1142.test_not_106 $106$ does not satisfy the Erdős 1142 property ($106 - 2 = 104 = 8 \times 13$)
Erdos1145.Erdos1145Prop Let $A=\{1\leq a_1 < a_2 < \cdots\}$ and $B=\{1\leq b_1 < b_2 < \cdots\}$ be sets of integers with $a_n/b_n\to 1$
Erdos1145.erdos_1145 Let $A=\{1\leq a_1 < a_2 < \cdots\}$ and $B=\{1\leq b_1 < b_2 < \cdots\}$ be sets of integers with $a_n/b_n\to 1$
Erdos1145.erdos_1145.test_implies_erdos_28 A stronger form of [erdosproblems.com/28]
Erdos1146.IsEssentialComponent We say that $A\subset \mathbb{N}$ is an essential component if $d_s(A \oplus B)>d_s(B)$ for every $B\subset \mathbb{N}$ with $0<d_s(B)<1$ where $d_s$ is the Schnirelmann density
Erdos1146.erdos_1146 Is $B=\{2^m3^n : m,n\geq 0\}$ an essential component?
Erdos1147.recurrenceSet The set $\{ n\geq 1: \| \alpha n^2\| < \epsilon(n)\}$
Erdos1147.erdos_1147.variants.almost_every Documented declaration: almost_every
Erdos1148.Erdos1148Prop A natural number $n$ which can be written as $n$ if $n = x^2 + y^2 - z^2$ with $\max(x^2, y^2, z^2) \leq n$
Erdos1148.erdos_1148.variants.lower_bound The integer $6563$ cannot be written as $x^2 + y^2 - z^2$ with $\max(x^2, y^2, z^2) \leq 6563$
Erdos1148.erdos_1148_weaker_prop The weaker property: $n = x^2 + y^2 - z^2$ such that $\max(x^2, y^2, z^2) \leq n + 2\sqrt{n}$
Erdos1148.erdos_1148.variants.weaker [Va99] reports this is 'obvious' if we replace $\leq n$ with $\leq n+2\sqrt{n}$
Erdos115.erdos_115 Documented declaration: erdos_115
Erdos1150.erdos_1150 Is there some constant
Erdos1150.erdos_1150.variants.parseval_lower_bound The trivial lower bound from Parseval's identity: for any polynomial $P$ of degree $n$ with coefficients in $\{-1, 1\}$, we have $\max_{|z|=1} |P(z)| \geq \sqrt{n+1}$
Erdos1157.erdos_1157 Let $s,k,r\geq 2$
Erdos1158.erdos_1158 Let $K_{t}(r)$ be the complete $t$-partite $t$-uniform hypergraph with $r$ vertices in each class
Erdos1159.erdos_1159 Determine whether there exists a constant $C>1$ such that the following holds
Erdos116.erdos_116.variants.log The lower bound
Erdos116.erdos_116.variants.upper Documented declaration: upper
Erdos116.erdos_116.variants.polya Pólya [Po28]: $\lvert\{ z: \lvert p(z)\rvert <1\}\rvert \leq \pi$ always holds
Erdos1167.erdos_1167 Erdős Problem 1167
Erdos1167.erdos_1167.variants.finite_targets Finite-target case
Erdos1167.erdos_1167.variants.binary_colors Binary-color case
Erdos1167.erdos_1167.variants.infinite_targets Infinite-target case
Erdos1167.erdos_1167.variants.r_eq_two $r = 2$ case
Erdos1167.i0 A canonical element of the type (1 : Ordinal).ToType
Erdos1167.cardinalPartitionRel_one The partition relation $\mu \to (\nu)^r_1$ with a single color is equivalent to $\nu \le \mu$
Erdos1167.erdos_1167.unrestricted_is_false The unrestricted version of Erdős Problem 1167 (without the condition $\gamma \geq 2$ from the original Erdős–Hajnal list) is false
Erdos1167.erdos_1167.variants.infinite_targets_needs_bound The infinite_targets variant without the bound $\kappa_\alpha \leq \lambda$ is false
Erdos1175.erdos_1175 Let $\kappa$ be an uncountable cardinal
Erdos1175.erdos_1175.variants.aleph_one The case $\kappa = \lambda = \aleph_1$ of Problem 1175: does every graph with chromatic number $\aleph_1$ contain a triangle-free subgraph with chromatic number $\aleph_1$?
Erdos1175.erdos_1175.variants.threshold_formulation Threshold reformulation variant
Erdos1175.erdos_1175.test.exists_triangle_free_subgraph Documented declaration: exists_triangle_free_subgraph
Erdos1175.erdos_1175.test.threshold_implies_exact Documented declaration: threshold_implies_exact
Erdos1176.erdos_1176 Let $G$ be a graph with chromatic number $\aleph_1$
Erdos1177.erdos_1177.parts.i Let $G$ be a finite $3$-uniform hypergraph, and let $F_G(\kappa)$ denote the collection of $3$-uniform hypergraphs with chromatic number $\kappa$ not containing $G$
Erdos1177.erdos_1177.parts.ii If both $F_G(\aleph_1)$ and $F_H(\aleph_1)$ are non-empty then $F_G(\aleph_1)\cap F_H(\aleph_1)$ is non-empty
Erdos1177.erdos_1177.parts.iii If $\kappa,\lambda$ are uncountable cardinals and $F_G(\kappa)$ is non-empty then $F_G(\lambda)$ is non-empty
Erdos1178.erdos_1178 Documented declaration: erdos_1178
Erdos1179.repCount $F_A(g) = \#\{ S\subseteq A : g = \sum_{x\in S}x\}$
Erdos1179.IsBalanced A is ε-balanced: $\lvert F_A(g)-2^{|A|}/N\rvert \leq \epsilon 2^{|A|}/N$ for all g
Erdos1179.successProbability The probability that a uniformly random k-element subset of G is ε-balanced
Erdos1180.Represents Represents ε p a s says that the multiset s of denominators n with 1 ≤ n ≤ p^ε and n coprime to p has ∑ n⁻¹ = a in ZMod p
Erdos1180.C C ε is the least C such that, for all primes p, every residue modulo p is the sum of at most C elements of {n⁻¹ : 1 ≤ n ≤ p^ε} (or 0 if no such C exists)
Erdos1180.erdos_1180.variants.shparlinski Shparlinski [Sh02] proved that $\ll \epsilon^{-3}$ summands suffice for every residue modulo every sufficiently large prime, where the threshold on $p$ may depend on $\epsilon$
Erdos1180.erdos_1180.variants.glibichuk Glibichuk [Gl06] improved the bound to $\ll \epsilon^{-2}$ summands, again for every sufficiently large prime
Erdos1180.erdos_1180.variants.lower_bound It is trivial that at least $\gg \epsilon^{-1}$ summands are required
Erdos1180.erdos_1180.variants.optimal Is $C_\epsilon\leq \epsilon^{-1-o(1)}$?
Erdos1188.CongruenceClass A congruence class (n, a) represents a (mod n)
Erdos1188.ValidClass The modulus is > 1 and the residue is reduced
Erdos1188.Satisfies z lies in the class (n, a)
Erdos1188.Covers Every integer satisfies some congruence of S
Erdos1188.HasDistinctModuli No two classes of S share a modulus
Erdos1188.IsMinimalDistinctCoveringSystem A minimal distinct covering system: valid classes with distinct moduli that cover ℤ, with no proper subset already covering
Erdos1188.ClassesUpTo All classes (n, a) with 2 ≤ n ≤ x and a < n
Erdos1188.coveringCount F(x): the number of minimal distinct covering systems all of whose moduli lie in [1, x]
Erdos1188.erdos_1188 Call a set of distinct integers
Erdos119.p Let $z_i$ be an infinite sequence of complex numbers such that $|z_i| = 1$ for all $i \geq 1$
Erdos119.M Let $M_n = \max_{|z| = 1} |p_n(z)|$
Erdos119.erdos_119.parts.i Is it true that $\limsup M_n = \infty$?
Erdos119.erdos_119.parts.ii Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_n > n^c$?
Erdos119.erdos_119.parts.iii Is it true that there exists $c > 0$ such that, for all large $n$, $\sum_{k \leq n} M_k > n^{1 + c}$?
Erdos1190.eps Documented declaration: eps
Erdos1190.erdos_1190 Documented declaration: erdos_1190
Erdos1190.erdos_1190.variants.lt_one Erdős [Er80] seems to credit Mirsky and Newman with the result that $\epsilon_m<1$, but gives no reference
Erdos1190.erdos_1190.variants.tendsto_zero He could not even decide whether $\epsilon_m\to 0$ as $m\to \infty$
Erdos1190.erdos_1190.variants.lower_bound The work of de la Bretèche, Ford, and Vandehey [BFV13] implies $$L(m)^{-1+o(1)}< \epsilon_m < L(m)^{-\sqrt{3}/2+o(1)},$$ where $L(m)=\exp(\sqrt{\log m\log\log m})$
Erdos1190.erdos_1190.variants.upper_bound The work of de la Bretèche, Ford, and Vandehey [BFV13] implies $$L(m)^{-1+o(1)}< \epsilon_m < L(m)^{-\sqrt{3}/2+o(1)},$$ where $L(m)=\exp(\sqrt{\log m\log\log m})$
Erdos1192.f_r For $A\subset \mathbb{N}$ let $f_r(n)$ count the number of solutions to $n=a_1+\cdots+a_r$ with $a_i\in A$
Erdos1192.erdos_1192.f_r_zero_zero The empty sum ($r = 0$) gives exactly one representation of $0$: the empty tuple
Erdos1192.erdos_1192.f_r_empty With an empty set, there are no valid $r$-tuples for $r \geq 1$
Erdos1192.erdos_1192.f_r_singleton_self For $r = 1$, $f_1(\{n\}, n) = 1$: the only $1$-tuple from $\{n\}$ summing to $n$ is $(n)$
Erdos1192.erdos_1192.f_r_no_rep $f_1(\{0\}, 1) = 0$: can't represent $1$ with a single element from $\{0\}$
Erdos1192.erdos_1192 Does there exist, for all $r\geq 2$, a basis $A$ of order $r$ (so that $f_r(n)>0$ for all large $n$) such that $$\sum_{n\leq x}f_r(n)^2 \ll x$$ for all $x$?
Erdos1192.erdos_1192.variants.renyi Erdős and Rényi proved by the probabilistic method that there exists a set $A$ such that $$\sum_{n\leq x}f_r(n)^2 \ll x$$ and $$\lvert A\cap [1,x]\rvert\gg x^{1/r}$$ for all $x$
Erdos1192.erdos_1192.variants.ruzsa Ruzsa [Ru90] proved that the answer is yes for $r=2$
Erdos1193.erdos_1193.parts.i Let $A\subset \mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$
Erdos1193.erdos_1193.parts.ii Let $A\subset \mathbb{N}$ and let $g(n)$ be a non-decreasing function of $n$ which is always $>0$
Erdos1193.erdos_1193.variants.sumRep_univ Indeed if $A=\mathbb{N}$ (assuming $0\in\mathbb{N}$) then $1_A\ast 1_A(n)=n+1$ for all $n$
Erdos1193.erdos_1193.variants.upper_density_pos Erdős writes the upper density can be positive, but he believes it is bounded away from $1$
Erdos1196.IsPrimitive A set is primitive if no non-associated elements of the set divide each other
Erdos1199.erdos_1199 Is it true that in any 2-colouring of $\mathbb{N}$ there exists an infinite set $A$ such that all elements of $A+A$ are the same colour?
Erdos12.IsGood A set A is "good" if it is infinite and there are no distinct a,b,c in A such that a ∣ (b+c) and b > a, c > a
Erdos12.erdos_12.parts.iii Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$
Erdos12.erdos_12.variants.erdos_sarkozy_density_0 Erdős and Sárközy proved that such an $A$ must have density 0
Erdos12.erdos_12.variants.erdos_sarkozy Given any function
Erdos12.erdos_12.variants.example Documented declaration: example
Erdos12.erdos_12.variants.schoen Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$
Erdos12.erdos_12.variants.baier Let $A$ be a set of natural numbers with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$
Erdos120.Erdos120For There exists a set
Erdos120.erdos_120 Let $A \subseteq \mathbb{R}$ be an infinite set
Erdos120.erdos_120.variants.finite_set Steinhaus [St20] has proved Erdős 120 to be false whenever $A$ is a finite set
Erdos1201.Erdos1201Set The set of $n$ for which $P(n(n+1)\cdots(n+k)) > n^{1-\epsilon}$, where $P(m)$ is the greatest prime divisor of $m$
Erdos1201.erdos_1201 Documented declaration: erdos_1201
Erdos1201.erdos_1201.variants.epsilon_half Erdős wrote he could prove this for $\epsilon=1/2$
Erdos1203.F If $\omega(n)$ counts the number of distinct prime divisors of $n$ then let $F(n)=\max_k \omega(n+k)\frac{\log\log k}{\log k}.$
Erdos1203.erdos_1203 Prove that $F(n)\to \infty$ as $n\to \infty$
Erdos1203.erdos_1203.variants.lower_bound It is easy to prove that $F(n)\geq 1-o(1)$
Erdos1206.erdos_1206.parts.i Does $\{1,2^3,\ldots,N^3\}$ contain a Sidon set of size $\gg N$?
Erdos1206.erdos_1206.parts.ii Is there an infinite set $A\subset \mathbb{N}$ of positive density such that $\{a^3 : a\in A\}$ is a Sidon set?
Erdos1207.P P d n is the largest number $m$ such that every set of $n$ points in $\mathbb{R}^d$ has an isosceles-free subset of size at least $m$
Erdos1207.erdos_1207 Let $P_d(n)$ be such that in any set of $n$ points in $\mathbb{R}^d$ there exist at least $P_d(n)$ many points which do not contain an isosceles triangle
Erdos1209.exists_prime_gt_not_prime_add For every k ≥ 1 and bound m, there is a prime p > m such that k + p is composite: take a prime q > k and, by Dirichlet, a prime p > max m q with p ≡ -k (mod q)
Erdos1209.seq The counterexample sequence: a 0 is a prime at least f 0, and a (k + 1) is a prime larger than a k and f (k + 1) with (k + 1) + a (k + 1) composite
Erdos1209.seq_zero_spec The defining properties of seq f 0
Erdos1209.seq_succ_spec The defining properties of seq f (k + 1)
Erdos1209.erdos_1209.parts.i Let $A=\{a_1a_{k-1}$ be a prime such that $a_k+k\equiv 0\pmod{q_k}$, where $q_k$ is some prime not dividing $k$
Erdos1209.exists_prime_gt_not_squarefree_add For every k ≥ 1 and bound m, there is a prime p > m such that k + p is not squarefree: take a prime q > k and, by Dirichlet, a prime p > m with p ≡ -k (mod q ^ 2)
Erdos1209.seq' The counterexample sequence for the squarefree variant
Erdos1209.seq'_succ_spec The defining properties of seq' f (k + 1)
Erdos1209.erdos_1209.parts.ii What if we ask for $n+a_k$ to be squarefree instead of prime?
Erdos1209.erdos_1209.parts.iii.a Are there $n$ such that $n+2^{2^k}$ is always a prime?
Erdos1209.erdos_1209.parts.iii.b Are there $n$ such that $n+2^{2^k}$ is always squarefree?
Erdos1209.erdos_1209.parts.iii.c Are there $n$ such that $n+2^{2^k}$ is infinitely often a prime?
Erdos1209.erdos_1209.parts.iii.d Are there $n$ such that $n+2^{2^k}$ is infinitely often squarefree?
Erdos121.F $F_k(N)$ is the size of the largest $A\subseteq \{1,\ldots,N\}$ such that the product of no $k$ many distinct elements of $A$ is a square
Erdos121.erdos_121.variants.three Erdős, Sós, and Sárközy [ESS95] proved $F_3(N) = (1-o(1))N$
Erdos121.erdos_121.variants.two Erdős, Sós, and Sárközy [ESS95] proved $F_2(N)=\left(\frac{6}{\pi^2}+o(1)\right)N$
Erdos1210.erdos_1210 Let $A\subseteq [1,n)$ be a set of integers such that $(a,b)=1$ for all distinct $a,b\in A$
Erdos1210.erdos_1210.variants.er80_correction In [Er80] he claims he "did not state this quite correctly" in [Er77c]
Erdos1212.Valid A vertex of the strengthened problem: a visible lattice point with both coordinates exceeding 1 and at least one coordinate composite
Erdos1212.valid_four_three Sanity check for Valid: the vertex $(4, 3)$ is valid — both coordinates exceed $1$, they are coprime, and $4$ is composite
Erdos1212.Adj Two lattice points are adjacent iff they differ by exactly 1 in exactly one coordinate
Erdos1212.erdos_1212 Documented declaration: erdos_1212
Erdos1212.vertical_leg_valid Core of the composite-anchor reduction: vertical-leg vertices
Erdos1212.horizontal_leg_valid Core of the composite-anchor reduction: horizontal-leg vertices $(s, c)$ for $a \le s \le b$ are valid, given the anchor $c$ is composite and coprime to the whole leg
Erdos1212.anchor_coprime_of_short_leg Roughness criterion (sufficiency for the anchor conditions): if $a < s$ for all $s$ in the leg and the leg stays below $a + P^-(a)$, then $a$ is coprime to the whole leg
Erdos1212.right_neighbor_witness_free Documented declaration: right_neighbor_witness_free
Erdos1212.left_neighbor_witness_free Isolation lemma, left neighbour
Erdos1212.vertical_neighbor_both_even Isolation lemma, vertical neighbours: both coordinates even
Erdos1213.HasEqualIntervalSums A finite sequence A 0, …, A (s - 1) has two distinct (nonempty) index intervals [u, v) and [x, y) with the same sum
Erdos1213.erdos_1213.variants.hegyvari Hegyvári [He86] proved that one can take $f(a,K) \ll ae^{O(K)}$
Erdos1219.erdos_1219.test.aleph_omega_lt_sum Under the hypotheses of erdos_1219, the target $\aleph_\omega$ is smaller than the source $\sum_k 2^{\aleph_{n_k}}$, so the partition relation is not ruled out by cardinality
Erdos123.IsDComplete A sequence is said to be $d$-complete if every large integer is the sum of distinct integers from the sequence, none of which divide any other
Erdos123.IsSnug Characterizes a "snug" finite set of natural numbers: all elements are within a multiplicative factor $(1 + ε)$ of the minimum
Erdos123.PairwiseCoprime Predicate for pairwise coprimality of three integers
Erdos123.erdos_123.variants.erdos_lewin_3_5_7 Erdős and Lewin [ErLe96] proved this conjecture when $a = 3$, $b = 5$, and $c = 7$
Erdos123.erdos_123.variants.powers_2_3 A simpler case: the set of numbers of the form $2^k 3^l$ ($k, l ≥ 0$) is d-complete
Erdos123.erdos_123.variants.powers_2_3_5_snug In [Er92b] Erdős makes the stronger conjecture (for
Erdos124.sumsOfDistinctPowers The set of integers which are the sum of distinct powers d ^ i with i ≥ k
Erdos124.erdos124.zero Documented declaration: zero
Erdos124.erdos124.ne_zero Documented declaration: ne_zero
Erdos124.erdos124.ne_zero_three_four_seven Documented declaration: ne_zero_three_four_seven
Erdos124.erdos124.melfi_construction Documented declaration: melfi_construction
Erdos125.erdos_125.variants.positive_upper_density Literature question: Does $A + B$ have positive upper density?
Erdos125.erdos_125.variants.zero_density Case 1: Does $A + B$ have zero upper and lower density?
Erdos125.erdos_125.variants.zero_lower_positive_upper_density Case 2: Does $A + B$ have zero lower density, but positive upper density?
Erdos126.erdos_126.variants.IsBigO Documented declaration: IsBigO
Erdos126.erdos_126.variants.isLittleO Erdős says that $f(n) = o(\frac{n}{\log n})$ has never been proved
Erdos127.f f m is the largest k such that every graph with m edges contains a bipartite subgraph with at least $\frac{m}{2}+\frac{\sqrt{8m+1}-1}{8}+k$ edges
Erdos127.erdos_127.variants.alon_upper Alon [Al96] showed that $f(m)\ll m^{1/4}$ for all $m$
Erdos128.erdos_128 Let G be a graph with n vertices such that every induced subgraph on ≥ $n/2$ vertices has more than $n^2/50$ edges
Erdos13.IsForbiddenTripleFree A finite set of naturals A is said to be forbidden-triple-free if for all a, b, c ∈ A, if a < min(b, c) then a does not divide b + c
Erdos13.erdos_13.variants.general A general version asks, for a fixed
Erdos130.erdos_130 Let $A\subset\mathbb{R}^2$ be an infinite set which contains no three points on a line and no four points on a circle
Erdos131.NonDividing A finite set $A$ of naturals is non-dividing if no $a \in A$ divides the sum of the elements of any nonempty subset $S \subseteq A \setminus \{a\}$
Erdos131.F $F(N)$ is the maximal size of a non-dividing subset of $\{1, \dots, N\}$
Erdos131.exists_card_eq_F $F(N)$ is attained: some non-dividing $A \subseteq \{1, \dots, N\}$ has exactly $F(N)$ elements
Erdos131.card_le_F Every non-dividing $A \subseteq \{1, \dots, N\}$ has at most $F(N)$ elements
Erdos131.F_le Trivial upper bound: $F(N) \le N$
Erdos131.F_mono $F$ is monotone
Erdos131.one_le_F Trivial lower bound: for $N \ge 1$, the singleton $\{N\}$ is non-dividing, so $1 \le F(N)$
Erdos131.erdos_131.variants.pham_zakharov Every non-dividing set is non-averaging
Erdos131.erdos_131.variants.elrss Erdős, Lev, Rauzy, Sándor and Sárközy [ELRSS99] proved $F(N) < 3 N^{1/2} + 1$
Erdos131.erdos_131 Let $F(N)$ be the maximal size of $A \subseteq \{1, \dots, N\}$ such that no $a \in A$ divides the sum of any distinct elements of $A \setminus \{a\}$
Erdos133.f f n is the least possible maximum degree of a triangle-free graph on n vertices with diameter 2, i.e. the largest f such that every such graph has a vertex of degree ≥ f
Erdos133.erdos_133.variants.lower_bound The lower bound $f(n)\geq (1-o(1))\sqrt{n}$ follows from the fact that a graph with maximum degree $d$ and diameter $2$ has at most $d^2+1$ many vertices
Erdos133.erdos_133.variants.hanson_seyffarth Hanson and Seyffarth [HaSe84] proved that $f(n)\leq (\sqrt{2}+o(1))\sqrt{n}$
Erdos133.erdos_133.variants.furedi_seress Füredi and Seress [FuSe94] proved that $f(n)\leq (\frac{2}{\sqrt{3}}+o(1))\sqrt{n}$
Erdos133.erdos_133.variants.asymptotic Is $f(n)\sim \sqrt{n}$?
Erdos134.erdos_134 Let $\epsilon,\delta>0$ and $n$ be sufficiently large in terms of $\epsilon$ and $\delta$
Erdos136.Is45Coloring An edge colouring of K_n with k colours is a (4, 5)-colouring if the six edges of every copy of K_4 receive at least five distinct colours
Erdos136.Colorable K_n admits a (4, 5)-colouring with k colours
Erdos136.f f n is the least number of colours in a (4, 5)-colouring of K_n
Erdos137.erdos_137 We say that $N$ is powerful if whenever $p\mid N$ we also have $p^2\mid N$
Erdos137.erdos_137.variants.perfect_power Let $k\geq 2$
Erdos137.erdos_137.variants.multiple_powerful_factors Documented declaration: multiple_powerful_factors
Erdos138.monoAP_guarantee_set The set of natural numbers that guarantee a monochromatic arithmetic progression
Erdos138.monoAP_guarantee_set_nonempty Asserts that for any number of colors r and any progression length k, there always exists some number N large enough to guarantee a monochromatic arithmetic progression
Erdos138.monoAPNumber The van der Waerden number, is the smallest integer N such that any r-coloring of {1, ..., N} is guaranteed to contain a monochromatic arithmetic progression of length k
Erdos138.W An abbreviation for the van der Waerden number for 2 colors, commonly written as W(k)
Erdos138.erdos_138 In [Er80] Erdős asks whether $$ \lim_{k \to \infty} (W(k))^{1/k} = \infty $$
Erdos138.erdos_138.variants.prime When $p$ is prime Berlekamp [Be68] has proved $W(p+1) ≥ p^{2^p}$
Erdos138.erdos_138.variants.upper Gowers [Go01] has proved $$W(k) \leq 2^{2^{2^{2^{2^{k+9}}}}.$$
Erdos138.erdos_138.variants.quotient In [Er81] Erdős asks whether $\frac{W(k+1)}{W(k)} \to \infty$
Erdos14.nonUniqueSumCount Documented declaration: nonUniqueSumCount
Erdos14.erdos_14.parts.i Let $A ⊆ \mathbb{N}$
Erdos14.erdos_14.parts.ii Is it possible that $|\{1,\ldots,N\} \setminus B| = o(N^\frac{1}{2})$?
Erdos141.Set.IsPrimeProgressionOfLength The predicate that a set s consists of l consecutive primes (possibly infinite)
Erdos141.first_three_odd_primes The first three odd primes are an example of three consecutive primes
Erdos141.Set.IsAPAndPrimeProgressionOfLength The predicate that a set s is both an arithmetic progression of length l and a progression of l consecutive primes
Erdos141.exists_three_consecutive_primes_in_ap There are 3 consecutive primes in arithmetic progression
Erdos141.erdos_141 Let $k≥3$
Erdos141.erdos_141.variants.first_cases The existence of such progressions has been verified for $k≤10$
Erdos141.erdos_141.variants.eleven Are there $11$ consecutive primes in arithmetic progression?
Erdos141.consecutivePrimeArithmeticProgressions The set of arithmetic progressions of consecutive primes of length $k$
Erdos141.erdos_141.variants.infinite_three It is open, even for $k=3$, whether there are infinitely many such progressions
Erdos141.erdos_141.variants.infinite_general_case Fix a $k \geq 3$
Erdos142.erdos_142 Prove an asymptotic formula for $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression
Erdos142.erdos_142.variants.lower Documented declaration: lower
Erdos142.erdos_142.variants.isTheta Determine the order of magnitude of $r_k(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression
Erdos142.erdos_142.variants.three Prove an asymptotic formula for $r_3(N)$, the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $3$-term arithmetic progression
Erdos143.WellSeparatedSet Let $A \subseteq (1, \infty)$ be a countably infinite set such that for all $x\neq y\in A$ and integers $k \geq 1$ we have $|kx - y| \geq 1$
Erdos143.erdos_143.parts.i Does this imply that $$ \liminf \frac{|A \cap [1,x]|}{x} = 0? $$
Erdos143.erdos_143.parts.ii Or $$ \sum_{x \in A} \frac{1}{x \log x} < \infty, $$
Erdos145.s Let $s_1 < s_2 < \cdots$ be the sequence of squarefree numbers
Erdos145.A Let $A(x)$ denote the set of indices $n$ for which $s_n \leq x$
Erdos145.erdos_145 Let $s_1 < s_2 < \cdots$ be the sequence of squarefree numbers
Erdos145.erdos_145.variants.le_two Erdős [Er51] proved this for all $0\leq \alpha\leq 2$
Erdos145.erdos_145.variants.le_three Hooley [Ho73] extended this to all $0 \leq \alpha\leq 3$
Erdos145.erdos_145.variants.lt_eleven_thirds Greaves, Harman, and Huxley showed (in Chapter 11 of [GHH97]) that this is true for $0 \leq \alpha < 11/3$
Erdos146.erdos_146 If $H$ is bipartite and is $r$-degenerate, that is, every induced subgraph of $H$ has minimum degree $\leq r$, then $$\mathrm{ex}(n;H) \ll n^{2-1/r}.$$ The answer is no
Erdos147.erdos_147.variants.janzer_even Janzer [Ja23] disproved the conjecture for even
Erdos147.erdos_147.variants.cubic Janzer [Ja23b] constructed, for any $\epsilon>0$, a nonempty $3$-regular bipartite graph $H$ such that $\mathrm{ex}(n;H)\ll n^{\frac{4}{3}+\epsilon}$
Erdos147.erdos_147.variants.janzer_conjecture Janzer [Ja23] conjectures that for any $r\geq 3$ and $\epsilon>0$ there exists a nonempty $r$-regular graph $H$ such that $\mathrm{ex}(n;H) \ll n^{2-\frac{2}{r}+\epsilon}$
Erdos148.F Documented declaration: F
Erdos148.u The sequence $u_0 = 1$, $u_{n+1} = u_n(u_n + 1)$ of [ElPl21, Corollary 3]: $1, 2, 6, 42, 1806, \ldots$, a shifted copy of Sylvester's sequence
Erdos148.c₀ The constant $c_0 = \lim_{n \to \infty} u_n^{2^{-n}} = 1.5979102\ldots$ of [ElPl21, Corollary 3]
Erdos148.F_one The only representation of $1$ as a single unit fraction is $1 = \frac{1}{1}$
Erdos148.F_two $1$ is not a sum of two distinct unit fractions
Erdos148.eq_two_three_six If $0 < x < y < z$ and $1 = \frac{1}{x} + \frac{1}{y} + \frac{1}{z}$, then $(x, y, z) = (2, 3, 6)$
Erdos148.F_three The only representation of $1$ as a sum of three distinct unit fractions is $1 = \frac{1}{2} + \frac{1}{3} + \frac{1}{6}$
Erdos148.F_first_values The values $F(1), \ldots, F(8)$ are $1, 0, 1, 6, 72, 2320, 245765, 151182379$ (OEIS A006585)
Erdos148.u_first_values The first values of the sequence are $1, 2, 6, 42, 1806$
Erdos148.erdos_148 Let $F(k)$ be the number of solutions to $$ 1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},$$ where $1\leq n_1<\cdots<n_k$ are distinct integers
Erdos148.erdos_148.variants.lower_bound The current best bounds known are
Erdos148.erdos_148.variants.upper_bound The current best bounds known are
Erdos15.erdos_15 Is it true that $\sum_{n=1}^\infty(-1)^n\frac{n}{p_n}$ converges, where $p_n$ is the sequence of primes?
Erdos150.IsMinimalCut Documented declaration: IsMinimalCut
Erdos150.maxMinimalCuts The maximum number $c(n)$ of minimal cuts a graph on $n$ vertices can have
Erdos150.erdos_150.variants.erdos_nesetril Asked by Erdős and Nešetřil, who also ask whether $c(3m+2)=3^m$
Erdos150.erdos_150.variants.seymour Seymour observed that $c(3m+2)\geq 3^m$, as seen by the graph of $m$ independent paths of length $4$ joining two vertices
Erdos150.erdos_150.variants.lower_bound The current best-known bounds on $\alpha$ are $$1.4457\leq \alpha \leq \frac{1+\sqrt{5}}{2}\approx 1.618.$$ The lower bound is due to Gaspers and Mackenzie [GaMa18]
Erdos150.erdos_150.variants.upper_bound Documented declaration: upper_bound
Erdos152.f Define f n to be the minimum of |{s | s - 1 ∉ A + A, s ∈ A + A, s + 1 ∉ A + A}| as A ranges over all Sidon sets of size n
Erdos153.f Define $f(n)$ to be the minimum of $\frac{1}{t}\sum_{1\leq i<t}(s_{i+1}-s_i)^2$ as $A$ ranges over all Sidon sets of size $n$, where $A+A=\{s_1<\cdots<s_t\}$
Erdos153.erdos_153 Let $A$ be a finite Sidon set and $A+A=\{s_1<\cdots<s_t\}$
Erdos154.erdos_154.variants.lindstrom Documented declaration: lindstrom
Erdos155.F Let $F(N)$ be the size of the largest Sidon subset of $\{1, \dots, N\}$
Erdos155.erdos_155 Is it true that for every $k \geq 1$ we have $$ F(N + k) \leq F(N) + 1 $$ for all sufficiently large $N$?
Erdos156.minMaximalSidonSet The size of the smallest maximal Sidon set in $\{1, \dots, N\}$
Erdos156.erdos_156 Does there exist a maximal Sidon set $A\subset \{1,\ldots,N\}$ of size $O(N^{1/3})$?
Erdos156.erdos_156.variants.greedy_lower_bound It is easy to prove that the greedy construction of a maximal Sidon set in $\{1,\ldots,N\}$ has size $\gg N^{1/3}$
Erdos156.erdos_156.variants.ruzsa_upper_bound Ruzsa [Ru98b] constructed a maximal Sidon set of size $\ll (N\log N)^{1/3}$
Erdos158.B2 A set A ⊆ ℕ is said to be a B₂[g] set if for all n, the equation a + a' = n, a ≤ a', a, a' ∈ A has at most g solutions
Erdos158.b2_one A set is B₂[1] iff it is Sidon
Erdos158.erdos_158 Let A be an infinite B₂[2] set
Erdos158.erdos_158.variants.isSidon' Let A be an infinite Sidon set
Erdos158.erdos_158.variants.isSidon As a corollary of erdos_158.isSidon', we can prove that liminf |A ∩ {1, ..., N}| * N ^ (- 1 / 2) = 0 for any infinite Sidon set A
Erdos159.erdos_159 There exists some constant $c>0$ such that $$R(C_4,K_n) \ll n^{2-c}.$$ The prize of $100 is offered in [Er78] for a proof or disproof
Erdos16.Erdos16Set The set of odd integers not of the form $2^k+p$
Erdos16.density_zero A set of natural numbers has density 0
Erdos16.erdos_16 Is the set of odd integers not of the form $2^k+p$ the union of an infinite arithmetic progression and a set of density $0$?
Erdos16.positive_lower_density A set of natural numbers has positive lower density
Erdos16.erdos_16.variant.romanoff Romanoff [Ro34] showed that the set of odd integers of this form has positive density
Erdos16.erdos_16.variant.erdos Using covering congruences Erdős [Er50] proved that the set of odd integers which are not of this form contains an infinite arithmetic progression
Erdos160.erdos_160.h Documented declaration: h
Erdos160.erdos_160.known_upper On Mathoverflow user leechlattice shows that $h(n) \ll n^{\frac 2 3}$
Erdos160.erdos_160.better_upper Estimate $h(n)$ by finding a better upper bound
Erdos160.erdos_160.better_lower Estimate $h(n)$ by finding a better lower bound
Erdos160.erdos_160.variants.known_lower The observation of Zachary Hunter in that question coupled with the bounds of Kelley-Meka KeMe23 imply that $$h(N) \gg \exp(c(\log N)^{\frac 1 {12}})$$ for some $c > 0$
Erdos161.erdos_161.parts.i Let $\alpha\in[0,1/2)$ and $n,t\geq 1$
Erdos161.erdos_161.parts.ii For fixed $t\geq 3$, do all positive densities below $1/2$ give the same asymptotic order?
Erdos162.colourEdgeCount The number of edges of $K_n$ inside H that get colour b under the colouring c
Erdos162.IsBalancedAbove Every set H of at least k vertices contains more than $\alpha\binom{|H|}{2}$ edges of each colour under c
Erdos162.F $F(n, \alpha)$: the smallest k such that some 2-colouring of $K_n$ satisfies IsBalancedAbove n α c k
Erdos162.erdos_162 Let $\alpha > 0$ and $n \geq 1$
Erdos162.erdos_162.variants.bounds It is easy to show with the probabilistic method that there exist $c_1(\alpha), c_2(\alpha)$ such that $c_1(\alpha)\log n < F(n,\alpha) < c_2(\alpha)\log n$
Erdos162.erdos_162.variants.half At $\alpha = 1/2$ the two colour counts in $H$ add up to $\binom{|H|}{2}$, so both cannot exceed half of it
Erdos162.isBalancedAbove_succ k = n + 1 always works, since no vertex set has more than n elements
Erdos162.F_le_succ $F(n, \alpha)$ is at most n + 1
Erdos164.erdos_164 A set $A\subset \mathbb{N}$ is primitive if no member of $A$ divides another
Erdos166.erdos_166.variants.mattheus_verstraete Mattheus and Verstraëte [MaVe23] proved the explicit lower bound $$R(4,k) \gg \frac{k^3}{(\log k)^4}.$$
Erdos166.erdos_166.variants.mattheus_verstraete_implies The bound of Mattheus and Verstraëte answers erdos_166, with logarithmic exponent $4$
Erdos168.NonTernary Say a finite set of natural numbers is non ternary if it contains no 3-term arithmetic progression of the form n, 2n, 3n
Erdos168.IntervalNonTernarySets IntervalNonTernarySets N is the (fin)set of non ternary subsets of {1,...,N}
Erdos168.F F N is the size of the largest non ternary subset of {1,...,N}
Erdos168.mem_IntervalNonTernarySets_iff Sanity check: elements of IntervalNonTernarySets N are precisely non ternary subsets of {1,...,N}
Erdos168.F_eq_card Sanity check: if S is a maximal non ternary subset of {1,..., N} then F N is given by the cardinality of S
Erdos168.erdos_168.parts.i What is the limit $F(N)/N$ as $N \to \infty$?
Erdos168.erdos_168.parts.ii Is the limit $F(N)/N$ as $N \to \infty$ irrational?
Erdos168.erdos_168.variants.limit_exists The limit $F(N)/N$ as $N \to \infty$ exists
Erdos17.IsClusterPrime A prime $p$ is a cluster prime if every even natural number $n \le p - 3$ can be written as a difference of two primes $q_1 - q_2$ with $q_1, q_2 \le p$
Erdos17.erdos_17 Erdős Problem 17
Erdos17.clusterPrimeCount The counting function of cluster primes $\le n$
Erdos17.erdos_17.variants.upper_BES In 1999 Blecksmith, Erdős, and Selfridge [BES99] proved the upper bound $$\pi^{\mathcal{C}}(x) \ll_A x(\log x)^{-A}$$ for every real $A > 0$
Erdos17.erdos_17.variants.upper_Elsholtz In 2003, Elsholtz [El03] refined the upper bound to $$\pi^{\mathcal{C}}(x) \ll x\,\exp\!\bigl(-c(\log\log x)^2\bigr)$$ for every real $0 < c < 1/8$
Erdos17.isClusterPrime_97_isLeast_non_cluster $97$ is the smallest prime that is not a cluster prime
Erdos170.PerfectRuler Documented declaration: PerfectRuler
Erdos170.PerfectRulersLengthN We define the set of all $N$-perfect rulers $A$ of length $N$, i.e. subsets $A \subseteq \{0, \dots, N\}$, s.t
Erdos170.TrivialRuler The trivial ruler with all marks $\{0, \dots, N\}$
Erdos170.trivial_ruler_is_perfect Sanity Check: the trivial ruler is actually a perfect ruler if $K \geq N$
Erdos170.F We define a function F N as the minimum cardinality of an N-perfect ruler of length N
Erdos170.erdos170 The problem is to determine the limit of the sequence $\frac{F(N)}{\sqrt{N}}$ as $N \to \infty$
Erdos170.lower_bound A known lower bound to the limit by Leech [Le56], which is $1.56\dots$
Erdos170.upper_bound A known upper bound obtained by constructing Wichmann's Rulers [Wi63]
Erdos170.erdos170.existing_bounds The existence of the limit has been proved by Erdős and Gál [ErGa48]
Erdos172.erdos_172 Is it true that in any finite colouring of $\mathbb{N}$ there exist arbitrarily large finite $A$ such that all sums and products of distinct elements in $A$ are the same colour?
Erdos178.erdos_178 Let $A_1,A_2,\ldots$ be an infinite collection of infinite sets of integers, say $A_i=\{a_{i1}0$
Erdos18.practicalH For a practical number $n$, $h(n)$ is the maximum over all $1 ≤ m ≤ n$ of the minimum number of divisors of $n$ needed to represent $m$ as a sum of distinct divisors
Erdos18.practicalH_one $h(1) = 1$: we need the single divisor {1} to represent 1
Erdos18.practicalH_two $h(2) = 1$: divisors are {1, 2}, each of m=1,2 needs only 1 divisor
Erdos18.practicalH_six $h(6) = 2$: divisors are {1, 2, 3, 6}
Erdos18.practicalH_twelve $h(12) = 3$: divisors are {1, 2, 3, 4, 6, 12}
Erdos18.practicalH_le_divisors For any practical number $n$, $h(n)$ ≤ number of divisors of $n$
Erdos18.factorial_isPractical $h(n!)$ is well-defined since $n!$ is practical for $n ≥ 1$
Erdos18.erdos_18a Conjecture 1
Erdos18.erdos_18b Conjecture 2
Erdos18.erdos_18c Conjecture 3
Erdos18.erdos_18_upper_bound Erdős's Theorem
Erdos18.erdos_18_vose Vose's Theorem
Erdos180.FiniteGraph A finite graph, bundled with its vertex count, so that a family may mix orders
Erdos180.FamilyFree A host graph is family-free when it contains no member of family as a subgraph
Erdos180.familyExtremal $\mathrm{ex}(n;\mathcal{F})$, the greatest number of edges of a graph on n vertices containing no member of family
Erdos180.IsCyclicFamily No member of the family is acyclic
Erdos180.IsCompactFamily The family is compact: some single member already controls the family extremal number
Erdos180.erdos_180 Documented declaration: erdos_180
Erdos181.erdos_181 Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges)
Erdos183.ForcesMonochromaticTriangle n forces a monochromatic triangle on k colours when every k-colouring of the edges of K_n has a colour class containing a triangle
Erdos183.multicolourTriangleRamsey $R(3;k)$, the minimal n such that every k-colouring of the edges of K_n contains a monochromatic triangle
Erdos184.IsCycleOrEdge A graph $H$ is a cycle or an edge if it is connected and 2-regular, or if it has exactly one edge
Erdos184.erdos_184 Any graph on $n$ vertices can be decomposed into $O(n)$ many edge-disjoint cycles and edges
Erdos184.erdos_184.variants.n_log_n Erdős and Gallai [EGP66] proved that $O(n \log n)$ many cycles and edges suffices
Erdos184.erdos_184.variants.lower_bound The graph $K_{3,n-3}$ shows that at least $(1+c)n$ many cycles and edges are required, for some constant $c>0$
Erdos184.erdos_184.variants.covering In [Er71] Erdős suggests that only $n-1$ many cycles and edges are required if we do not require them to be edge-disjoint
Erdos184.erdos_184.variants.bucic_montgomery The best bound available is due to Bucić and Montgomery [BM22], who prove that $O(n\log^* n)$ many cycles and edges suffice, where $\log^*$ is the iterated logarithm function
Erdos184.erdos_184.variants.conlon_fox_sudakov Conlon, Fox, and Sudakov [CFS14] proved that $O_\epsilon(n)$ cycles and edges suffice if $G$ has minimum degree at least $\epsilon n$, for any $\epsilon>0$
Erdos185.f3 $f_3(n)$ is the maximal size of a subset of $\{0,1,2\}^n$ which contains no three points on a line (in $\mathbb{R}^n$)
Erdos185.erdos_185.variants.moser Moser showed that $f_3(n) \gg \frac{3^n}{\sqrt{n}}$
Erdos188.s The set of numbers
Erdos188.erdos_188 What is the smallest
Erdos188.erdos_188.variants.nonempty Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 14, 15): It has been shown that there is a large
Erdos188.erdos_188.variants.estimate Old and new problems and results in combinatorial number theory by Erdős & Graham (Page 15): How small can $M$ be made?
Erdos189.Erdos189For Erdős problem 189 asked whether the below holds for all rectangles
Erdos189.erdos_189 If $\mathbb{R}^2$ is finitely coloured then must there exist some colour class which contains the vertices of a rectangle of every area?
Erdos189.erdos_189.variants.square Graham claims this is "easy to see"
Erdos189.erdos_189.variants.parallelogram Seems to be open, as of January 2025
Erdos19.EFLConfig A configuration of n pairwise edge-disjoint copies of $K_n$ covering the vertex type V
Erdos19.EFLConfig.graph The union graph: distinct vertices are adjacent iff they lie in a common clique
Erdos19.EFLConfig.IsShared A vertex is shared if it lies in two different cliques
Erdos19.EFLConfig.sharedSet The finite set of shared vertices
Erdos19.EFLConfig.le_chromaticNumber The graph $G$ contains a copy of $K_n$, so $n \le \chi(G)$
Erdos19.EFLConfig.eq_of_not_shared A vertex that is not shared lies in exactly one clique
Erdos19.EFLConfig.colorable_of_sharedColoring If the shared vertices can be coloured with $n$ colours so that shared vertices in a common clique get distinct colours, then $G$ is $n$-colourable
Erdos19.EFLConfig.card_sharedSet_le There are at most $\binom{n}{2}$ shared vertices, since two cliques share at most one vertex
Erdos19.erdos_19 If $G$ is an edge-disjoint union of $n$ copies of $K_n$, then is $\chi(G) = n$?
Erdos19.erdos_19.variants.le_three The Erdős–Faber–Lovász conjecture holds for $n \le 3$
Erdos19.erdos_19.variants.lt_ten Hindman [Hi81] proved the Erdős–Faber–Lovász conjecture for $n < 10$
Erdos19.erdos_19.variants.kahn Kahn [Ka92] proved that $\chi(G) \le (1 + o(1))n$
Erdos19.erdos_19.variants.large_n Kang, Kelly, Kühn, Methuku and Osthus [KKKMO21] proved the Erdős–Faber–Lovász conjecture for all sufficiently large $n$
Erdos191.erdos_191.variants.conlon_fox_sudakov Documented declaration: conlon_fox_sudakov
Erdos191.erdos_191.variants.rodl_upper Rödl [Ro03] constructed, for all
Erdos191.erdos_191.variants.three_colours Rödl [Ro03] proved that the answer to erdős_191 is negative if we consider
Erdos192.IsPositiveUnitWalk A sequence $a_1, a_2, \ldots \in \mathbb{R}^d$ such that every difference $a_{i+1}-a_i$ is a positive unit vector $(0,\ldots,0,1,0,\ldots,0)$
Erdos193.IsSWalk An $S$-walk is a sequence where every difference is in $S$
Erdos193.HasCollinearTriple True if set $A$ contains 3 distinct collinear points over $R$
Erdos193.erdos_193_z2 [GeRa79] showed that the answer is yes for $\mathbb{Z}^2$
Erdos195.erdos_195 What is the largest $k$ such that in any permutation of $\mathbb{Z}$ there must exist a monotone $k$-term arithmetic progression $x_1 < \cdots < x_k$?
Erdos195.erdos_195.variants.leq_5_bound Geneson [Ge19] proved that k ≤ 5
Erdos195.erdos_195.variants.leq_4_bound Adenwalla [Ad22] proved that k ≤ 4
Erdos196.erdos_196 Must every permutation of $\mathbb{N}$, contain a monotone 4-term arithmetic progression?
Erdos197.erdos_197 Can $\mathbb{N}$ be partitioned into two sets, each of which can be permuted to avoid monotone 3-term arithmetic progressions?
Erdos198.baumgartner_strong Let $V$ be a vector space over the rationals and let $k$ be a fixed positive integer
Erdos198.baumgartner_headline The statement for which Baumgartner actually writes a proof
Erdos198.erdos_198.variants.concrete In fact one such sequence is $n! + n$
Erdos2.erdos_2 Can the smallest modulus of a covering system be arbitrarily large?
Erdos20.f Let $f(n,k)$ be minimal such that every $F$ family of $n$-uniform sets with $|F| \ge f(n,k)$ contains a $k$-sunflower
Erdos20.erdos_20 Is it true that $f(n,k) < c_k^n$ for some constant $c_k>0$ and for all $n > 0$?
Erdos200.longestPrimeArithmeticProgressions The length of the longest arithmetic progression of primes in $\{1,\ldots,n\}$
Erdos200.erdos_200 Does the longest arithmetic progression of primes in $\{1,\ldots,N\}$ have length $o(\log N)$?
Erdos200.erdos_200.variants.upper It follows from the prime number theorem that such a progression has length $\leq(1+o(1))\log N$
Erdos202.f f N is the maximum possible r such that there are moduli $n_1<\cdots<n_r\leq N$ with associated residues $a_i\pmod{n_i}$ whose congruence classes are disjoint
Erdos202.erdos_202 Documented declaration: erdos_202
Erdos202.erdos_202.variants.erdos_szemeredi Erdős and Stein conjectured that $f(N)=o(N)$, which was proved by Erdős and Szemerédi [ErSz68]
Erdos203.erdos_203 Is there an integer $m$ with $(m, 6) = 1$ such that none of $2^k \cdot 3^\ell \cdot m + 1$ are prime, for any $k, \ell \ge 0$?
Erdos204.erdos_204 Are there $n$ such that there is a covering system with moduli the divisors of $n$ which is 'as disjoint as possible'?
Erdos205.IsRepresentable IsRepresentable f n states that n can be written as 2 ^ k + m for some k ≥ 0 where the number of prime divisors of m, counted with multiplicity, is less than f m
Erdos205.erdos_205.parts.i Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$?
Erdos205.erdos_205.parts.ii Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$?
Erdos205.erdos_205.parts.iii Is it true that all sufficiently large $n$ can be written as $2^k+m$ for some $k\geq 0$, where $\Omega(m)<\log\log m$?
Erdos205.erdos_205.variants.many_prime_factors In fact there are infinitely many $n$ such that, for all $k$ with $2^k<n$, $n-2^k$ has at least $$\gg \left(\frac{\log n}{\log\log n}\right)^{1/2}$$ many prime factors
Erdos205.erdos_205.variants.odd_counterexamples The $n$ constructed in this way are divisible by a large power of $2$
Erdos206.egyptianSum The Egyptian fraction sum: ∑_{m ∈ S} 1/m for a finset of natural numbers
Erdos206.IsUnderapprox S is an Egyptian underapproximation of x: valid denominators (all positive) and sum < x
Erdos206.IsBestNTerm S achieves the best n-term Egyptian underapproximation of x, i.e. egyptianSum S realises $R_n(x)$
Erdos206.EventuallyGreedy Documented declaration: EventuallyGreedy
Erdos206.erdos_206 Let $x>0$ be a real number
Erdos207.erdos_207 Documented declaration: erdos_207
Erdos208.erdos208.s The sequence of squarefree numbers, denoted by s as in Erdős problem 208
Erdos208.erdos_208.parts.i Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers
Erdos208.erdos_208.parts.ii Let $s_1 < s_2 < \dots$ be the sequence of squarefree numbers
Erdos208.erdos_208.variants.log_bound In [Er79] Erdős says perhaps $s_{n+1} - s_n \ll \log s_n$, but he is 'very doubtful'
Erdos209.pointMultiplicity The number of lines from A that pass through the point p
Erdos209.HasGallaiTriangle Documented declaration: HasGallaiTriangle
Erdos209.erdos_209 Let $A$ be a finite collection of $d\geq 4$ non-parallel lines in $\mathbb{R}^2$ such that there are no points where at least four lines from $A$ meet
Erdos21.IsErdosLovaszFamily An Erdős–Lovász family of order n: an intersecting family of n-sets such that every set of size at most n - 1 is disjoint from some member
Erdos21.f f n is the minimal size of an Erdős–Lovász family of order n
Erdos21.erdos_21.variants.lower Erdős and Lovász [ErLo75] proved that $\frac{8}{3}n-3\leq f(n)$ for all $n$
Erdos21.erdos_21.variants.three_n It has been speculated (see e.g. [Ka94]) that the correct answer is $3n+O(1)$
Erdos21.erdos_21.variants.small_values $f(1)=1$, $f(2)=3$, $f(3)=6$ [Tr14], $f(4)=9$ [Tr14] and $f(5)=13$ [BaWa21]
Erdos212.erdos_212 Is there a dense subset of ℝ^2 such that all pairwise distances are rational?
Erdos213.Erdos213For The predicate (on $n$) that there exist $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, such that all pairwise distances are integers
Erdos213.erdos_213 Let $n \geq 4$
Erdos213.erdos_213.variants.KK08 The best construction to date, due to Kreisel and Kurz, has $n = 7$
Erdos214.unitSquare The four vertices of a unit square
Erdos214.UnitDistanceAvoiding Colouring the points of S blue and the points of Sᶜ red gives a red/blue colouring of $\mathbb{R}^2$; it is unit-distance-avoiding if no two blue points are distance $1$ apart
Erdos214.HasRedCopies Any unit-distance-avoiding colouring contains a congruent red copy of every set of n points
Erdos214.k The largest integer k such that any unit-distance-avoiding colouring contains a congruent red copy of every set of k points
Erdos214.erdos_214 Let $S\subset \mathbb{R}^2$ be such that no two points in $S$ are distance $1$ apart
Erdos214.erdos_214.variants.juhasz The answer is yes, proved by Juhász [Ju79], who proved more generally that the complement of $S$ must contain a congruent copy of any set of four points
Erdos214.erdos_214.variants.egmrss Erdős, Graham, Montgomery, Rothschild, Spencer, and Straus [EGMRSS75] had earlier showed that $3\leq k\leq 10^{12}$
Erdos214.erdos_214.variants.juhasz_twelve_points Juhász [Ju79] also improved the upper bound to $k\leq 11$
Erdos214.erdos_214.variants.csizmadia_toth The upper bound was improved further by Csizmadia and Tóth [CsTo94] to $k\leq 7$
Erdos214.erdos_214.variants.bounds The best known bounds currently are $$4\leq k\leq 7.$$
Erdos218.erdos_218.variants.le The set of indices $n$ for which a prime gap is followed by a larger or equal prime gap has a natural density of $\frac 1 2$
Erdos218.erdos_218.variants.ge The set of indices $n$ for which a prime gap is preceded by a larger or equal prime gap has a natural density of $\frac 1 2$
Erdos218.erdos_218.variants.infinite_equal_prime_gap There are infinitely many indices $n$ such that the prime gap at $n$ is equal to the prime gap at $n+1$
Erdos219.primeArithmeticProgressions The set of arithmetic progressions of primes
Erdos22.erdos_22.variants.szemeredi_upper The matching upper bound, due to Szemerédi [Sz72]: a $K_4$-free graph on $n$ vertices whose independence number is sublinear in $n$ has at most $(1/8 + o(1))n^2$ edges
Erdos22.erdos_22.variants.bollobas_erdos_lower Documented declaration: bollobas_erdos_lower
Erdos22.erdos_22.variants.fox_loh_zhao The quantitative strengthening proved by Fox, Loh, and Zhao [FLZ15]: there is a constant
Erdos22.erdos_22.variants.test_bot A sanity check for erdos_22.variants.szemeredi_upper: the empty graph is $K_4$-free and trivially satisfies the upper bound $(1/8 + \epsilon)n^2$ on the number of edges
Erdos220.sumSquaredGaps The sum of the squared differences of adjacent entries of a list of natural numbers
Erdos220.sortedTotatives The reduced residues 1 ≤ m < n with (m, n) = 1, in increasing order
Erdos221.erdos_221 Is there a set
Erdos224.ObtuseAt The angle $yxz$ is obtuse
Erdos225.trigPoly The trigonometric polynomial $f(z) = \sum_{0\leq k\leq n}c_k e^{ikz}$, as an entire function of $z\in\mathbb{C}$
Erdos226.PreservesRationality A real function preserves rationality
Erdos228.erdos_228 Does there exist, for all large
Erdos229.erdos_229 Let $(S_n)_{n \ge 1}$ be a sequence of sets of complex numbers, none of which have a finite limit point
Erdos229.theorem_1 Let $\{S_k\}$ be any sequence of sets in the complex plane, each of which has no finite limit point
Erdos23.erdos_23.variants.n1 Every triangle-free graph on $5$ vertices can be made bipartite by removing at most $1$ edge
Erdos23.erdos_23.variants.n1_tight There exists a triangle-free graph on $5$ vertices such that at least $1$ edge must be removed to make it bipartite
Erdos23.erdos_23.variants.n5 Every triangle-free graph on $25$ vertices can be made bipartite by removing at most $25$ edges
Erdos23.erdos_23.variants.n5_tight There exists a triangle-free graph on $25$ vertices such that at least $25$ edges must be removed to make it bipartite
Erdos23.blowupC5 The blow-up of the 5-cycle
Erdos23.blowupC5_tight The blow-up of $C_5$ shows that the bound $n^2$ in Erdős Problem 23 is tight: any bipartite subgraph must omit at least $n^2$ edges
Erdos23.erdos_23 Can every triangle-free graph on $5n$ vertices be made bipartite by deleting at most $n^2$ edges?
Erdos230.circleMax The maximum of $\lvert P(z)\rvert$ over the unit circle, for $P(z)=\sum_{1\leq k\leq n}a_kz^k$
Erdos230.erdos_230.variants.ultraflat Kahane [Ka80] constructed 'ultraflat' polynomials
Erdos231.ContainsAbelianSquare A string S contains an abelian square if it has two consecutive nonempty blocks x and y such that y is a permutation of x
Erdos231.containsAbelianSquare_iff S contains an abelian square iff two consecutive blocks of some positive length L of S are permutations of each other
Erdos231.not_containsAbelianSquare_example The string $1213121412132124$ of length $2^4$ over four characters contains no abelian square
Erdos231.containsAbelianSquare_of_length_four Every string of length $4$ over two characters contains an abelian square
Erdos231.containsAbelianSquare_of_length_eight Every string of length $8$ over three characters contains an abelian square
Erdos231.erdos_231.variants.two_pow Perhaps Erdős meant
Erdos231.erdos_231.variants.two_pow_small Every string of length $2^k$ over $k$ characters contains an abelian square for $k = 2, 3$
Erdos233.erdos_233 A conjecture by Heath-Brown: The sum of squares of the first $N$ gaps between consecutive primes behaves like $N * (log N)^2$
Erdos233.erdos_233.variants.upper_bound Cramér proved an upper bound of $O(N(\log N)^4)$ conditional on the Riemann hypothesis
Erdos233.erdos_233.variants.lower_bound The prime number theorem immediately implies a lower bound of $\gg N(\log N)^2$ for the sum of squares of gaps between consecutive primes
Erdos234.erdos_234 Is it true that for all c ≥ 0, the density f c of integers for which (p (n + 1) - p n) / log n < c exists and is a continuous function of c?
Erdos235.N The product of the first k + 1 primes
Erdos235.coprimeList The increasing list of integers < n that are coprime to n
Erdos235.gaps The gaps $a_i - a_{i-1}$ between consecutive entries of coprimeList n
Erdos235.countGaps The number of gaps of size at most $c \cdot n / \varphi(n)$
Erdos235.proportion The proportion appearing in erdos_235, for the primorial N k
Erdos235.N_zero N 0 is the primorial of $2$
Erdos235.N_one N 1 is the primorial of $3$
Erdos235.coprimeList_length coprimeList n has length $\varphi(n)$
Erdos235.totient_N_pos $\varphi(N_k)$ is positive, so the proportion is not a division by zero
Erdos236.f $f(n)$ counts the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$
Erdos236.erdos_236 Let $f(n)$ count the number of solutions to $n=p+2^k$ for prime $p$ and $k\geq 0$
Erdos238.erdos_238 Let c₁, c₂ > 0
Erdos238.erdos_238.variants.small_c1 It is well-known that the conjecture above is true when c₁ is sufficiently small
Erdos239.erdos_239 Let $f:\mathbb{N}\to \{-1,1\}$ be a multiplicative function
Erdos24.erdos_24 Does every triangle-free graph on $5n$ vertices contain at most $n^5$ copies of $C_5$?
Erdos241.f Let $f(N)$ be the maximum size of $A\subseteq \{1,\ldots,N\}$ such that the sums $a+b+c$ with $a,b,c\in A$ are all distinct (aside from the trivial coincidences)
Erdos241.erdos_241 Is it true that $f(N)\sim N^{1/3}$?
Erdos241.erdos_241.variants.lower_bound Bose and Chowla [BoCh62] provided a construction proving one half of this, namely $(1+o(1))N^{1/3}\leq f(N)$
Erdos241.erdos_241.variants.upper_bound Green [Gr01] proved the upper bound $f(N) \leq ((7/2)^{1/3}+o(1))N^{1/3}$ (note that $(7/2)^{1/3}\approx 1.519$)
Erdos241.erdos_241.variants.upper_bound_white Green's argument gives
Erdos241.BoseChowlaConjecture The conjecture that the size of the set $A\subseteq \{1,\ldots,N\}$ is asymptotically $N^{1/r}$
Erdos241.erdos_241.variants.generalization Documented declaration: generalization
Erdos241.erdos_241.variants.r_eq_2 This is known only for $r=2$ (see [erdosproblems.com/30])
Erdos242.erdos_242 For every $n>2$ there exist distinct integers $1 ≤ x < y < z$ such that $\frac 4 n = \frac 1 x + \frac 1 y + \frac 1 z$
Erdos242.erdos_242.variants.schinzel_generalization Schinzel conjectured (see [Si56]) the generalisation that, for any fixed
Erdos243.erdos_243 Let $a_1 < a_2 < \dots$ be a sequence of integers such that $\lim_{n\to\infty} \frac{a_n}{a_{n-1}^2} = 1$ and $\sum \frac{1}{a_n} \in \mathbb{Q}$
Erdos244.erdos_244 Let $C > 1$
Erdos244.erdos_244.variants.Romanoff Romanoff [Ro34] proved that the answer is yes if $C$ is an integer
Erdos245.erdos_245 Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$
Erdos245.erdos_245.variants.two Let $A\subseteq\mathbb{N}$ be an infinite set such that $|A\cap \{1, ..., N\}| = o(N)$
Erdos246.Gamma The set $\{a^k b^l : k,l \geq 0\}$
Erdos247.erdos_247 Let $n_1 < n_2 < \cdots$ be a sequence of integers such that $$ \limsup \frac{n_k}{k} = \infty. $$ Is $$ \sum_{k=1}^{\infty} \frac{1}{2^{n_k}} $$ transcendental?
Erdos247.erdos_247.variants.strong_condition Erdős proved the answer is yes under the stronger condition that $\limsup \frac{n_k}{k^t} = \infty$ for all $t\geq 1$
Erdos249.erdos_249 Is $$\sum_{n} \frac{\phi(n)}{2^n}$$ irrational?
Erdos25.erdos_25 Let $n_1 < n_2 < \dots$ be an arbitrary sequence of integers, each with an associated residue class $a_i \pmod{n_i}$
Erdos251.erdos_251 Is $\sum_{n=1}^\infty \frac{p_n}{2^n}$ irrational?
Erdos252.erdos_252_sum The series ∑ σ k n / n!
Erdos252.erdos_252 Erdős Problem 252: irrationality of the sum for a given $k$
Erdos252.erdos_252.variants.k_eq_zero ∑ σ 0 n / n! is irrational
Erdos252.erdos_252.variants.k_eq_one ∑ σ 1 n / n! is irrational
Erdos252.erdos_252.variants.k_eq_two ∑ σ 2 n / n! is irrational
Erdos252.erdos_252.variants.k_eq_three ∑ σ 3 n / n! is irrational
Erdos252.erdos_252.variants.k_eq_four ∑ σ 4 n / n! is irrational
Erdos252.erdos_252.variants.k_ge_five For a fixed k ≥ 5, is ∑ σ k n / n! irrational?
Erdos252.erdos_252.variants.schinzel If Schinzel's conjecture is true, then ∑ σ k n / n! is irrational for all k
Erdos252.erdos_252.variants.prime_tuples If the prime k-tuples conjecture is true, then ∑ σ k n / n! is irrational
Erdos253.RepresentsAPs The predicate that a : ℕ → ℕ is a strictly monotone sequence such that every infinite arithmetic progression contains infinitely many integers that are the sum of distinct
Erdos253.erdos_253 Let $a_1 < a_2 < \dotsc$ be an infinite sequence of positive integers such that $\frac{a_{i+1}}{a_i} \to 1$
Erdos254.IsSumOfDistinct An integer n can be written as a sum of distinct elements of A
Erdos254.not_summable_iff_tendsto_partial_sums Documented declaration: not_summable_iff_tendsto_partial_sums
Erdos254.erdos_254 Documented declaration: erdos_254
Erdos254.erdos_254.variants.cassels Cassels [Ca60] proved this under the alternative hypotheses
Erdos257.erdos_257 Let $A\subseteq\mathbb{N}$ be an infinite set
Erdos257.erdos_257.variants.tsum_top_eq Show that $$ \sum_{n} \frac{1}{2^n - 1} = \sum_{n} \frac{d(n)}{2^n}, $$ where $d(n)$ is the number of divisors of $n$
Erdos258.erdos_258.variants.monotone Let $2 \leq a_1 \leq a_2 \leq \cdots$ be a monotone sequence with $a_n \to \infty$
Erdos259.erdos_259 Is $\sum_{n} \mu(n)^2\frac{n}{2^n}$ irrational?
Erdos26.IsThick A sequence of naturals $(a_i)$ is thick if their sum of reciprocals diverges: $$ \sum_i \frac{1}{a_i} = \infty $$
Erdos26.MultiplesOf The set of multiples of a sequence $(a_i)$ is $\{na_i | n \in \mathbb{N}, i\}$
Erdos26.IsBehrend A sequence of naturals $(a_i)$ is Behrend if almost all integers are a multiple of some $a_i$
Erdos26.IsWeaklyBehrend A sequence of naturals $(a_i)$ is weakly Behrend with respect to $\varepsilon \in \mathbb{R}$ if at least $1 - \varepsilon$ density of all numbers are a multiple of $A$
Erdos26.erdos_26 Let $A\subset\mathbb{N}$ be infinite such that $\sum_{a \in A} \frac{1}{a} = \infty$
Erdos261.Erdos261Prop A natural number
Erdos261.Erdos261InfiniteRepresentation A canonical infinite representation of a rational number $x$ by positive integers
Erdos261.erdos_261.variants.borwein_loring For every positive integer $m$, if $n = 2^{m+1} - m - 2$, then $$\frac{n}{2^n} = \sum_{n < k \le n + m} \frac{k}{2^k}.$$ This construction is due to Borwein and Loring [BoLo90]
Erdos261.erdos_261.variants.borwein_loring_property The Borwein--Loring construction gives the required property when $m \ge 2$
Erdos261.erdos_261.parts.i Are there infinitely many positive integers
Erdos261.erdos_261.variants.le_10000 Tengely, Ulas, and Zygadlo [TUZ20] verified that every positive integer $n \le 10000$ has the required property
Erdos261.erdos_261.parts.ii Do all positive integers $n$ have the required property?
Erdos261.erdos_261.parts.iii Is there a rational number $x$ such that $$x = \sum_{k=1}^{\infty} \frac{a_k}{2^{a_k}}$$ has at least $2^{\aleph_0}$ representations by pairwise distinct positive integers $a_k$?
Erdos263.IsIrrationalitySequence We call a strictly increasing sequence
Erdos263.IsWeakIrrationalitySequence Documented declaration: IsWeakIrrationalitySequence
Erdos263.erdos_263.parts.i Is $a_n = 2^{2^n}$ an irrationality sequence in the above sense?
Erdos263.erdos_263.parts.ii Must every irrationality sequence $a_n$ in the above sense satisfy $a_n^{1/n} \to \infty$ as $n \to \infty$?
Erdos263.erdos_263.variants.folklore A folklore result states that any $a_n$ satisfying $\lim_{n \to \infty} a_n^{\frac{1}{2^n}} = \infty$ has $\sum \frac{1}{a_n}$ converging to an irrational number
Erdos263.erdos_263.variants.sub_doubly_exponential Kovač and Tao [KoTa24] proved that any strictly increasing sequence
Erdos263.erdos_263.variants.super_doubly_exponential_eventual The same folklore result with the growth hypothesis stated as an eventual lower bound $a_{n+1} \geq c\, a_n^{2+\varepsilon}$
Erdos263.erdos_263.variants.doubly_exponential_all_but_countable Documented declaration: doubly_exponential_all_but_countable
Erdos264.IsIrrationalitySequence Documented declaration: IsIrrationalitySequence
Erdos264.erdos_264.parts.i Is $2^n$ an example of an irrationality sequence?
Erdos264.erdos_264.parts.ii Is $n!$ an example of an irrationality sequence?
Erdos264.erdos_264.variants.example One example is $2^{2^n}$
Erdos264.erdos_264.variants.ko_tao_neg Kovač and Tao [KoTa24] generally proved that any strictly increasing sequence of positive integers
Erdos264.erdos_264.variants.ko_tao_pos On the other hand, Kovač and Tao [KoTa24] do prove that for any function
Erdos266.erdos_266.variants.all_rationals In fact, Kovač and Tao proved in [KoTa24] that there exists a strictly increasing sequence
Erdos267.erdos_267 Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence
Erdos267.erdos_267.variants.generalisation_ratio_limit_to_infinity Let $F_1=F_2=1$ and $F_{n+1} = F_n + F_{n-1}$ be the Fibonacci sequence
Erdos267.erdos_267.variants.fibonacci_inverse_sum The sum $\sum_n \frac 1 {F_{n}}$ itself was proved to be irrational by André-Jeannin
Erdos269.HasPrimeFactorsIn A positive integer $n$ has all its prime factors in the set $P$
Erdos269.a The infinite, strictly increasing sequence $\{a_0, a_1, \dots\}$ of integers whose prime factors all belong to $P$
Erdos269.partialLcm The $n$-th partial least common multiple, $[a_0, \dots, a_{n-1}]$, which is the LCM of the first $n$ integers in the sequence
Erdos269.series The sum $\sum_{n=1}^\infty \frac{1}{[a_0,\ldots,a_{n - 1}]}$
Erdos269.erdos_269.variants.rational Let $P$ be a finite set of primes with $|P| \ge 2$ and let $\{a_1 < a_2 < \dots\}$ be the set of positive integers whose prime factors are all in $P$
Erdos269.erdos_269.variants.irrational Let $P$ be a finite set of primes with $|P| \ge 2$ and let $\{a_1 < a_2 < \dots\}$ be the set of positive integers whose prime factors are all in $P$
Erdos269.erdos_269.variants.infinite This theorem addresses the case where the set of primes $P$ is infinite
Erdos270.series The series $\sum_{n\geq 1} \frac{1}{(n+1)\cdots (n+f(n))}$, indexed from $0$
Erdos270.erdos_270.variants.monotone Documented declaration: monotone
Erdos270.erdos_270.variants.linear The case $f(n)=n$ has a positive answer, as observed by Crmarić and Kovač [Ko26]: $\sum_{n\geq 1} n!/(2n)!$ is irrational
Erdos272.IsArithInterSet Let $N \in\mathbb{N}$
Erdos272.maxArithInterCard For each $N > 0$, let $t$ be the largest size of an arithmetic intersection set
Erdos272.erdos_272 Let $N\geq 1$
Erdos272.erdos_272.variants.isBigO_sq Simonovits and Sós have shown that $t\ll N^2$
Erdos272.erdos_272.variants.szabo Szabo showed that the maximal $t$ is equal to $$ \frac{N^2}{2} + O(N^{5/3}\log^3N). $$
Erdos272.erdos_272.variants.szabo_strong Szabo asks whether the maximal $t$ is given by $$ \frac{N^2}{2} + O(N) $$
Erdos273.erdos_273 Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 5$?
Erdos273.erdos_273.variants.three Is there a covering system all of whose moduli are of the form $p-1$ for some primes $p \geq 3$?
Erdos274.Group.ExactCovering Documented declaration: ExactCovering
Erdos274.erdos_274 If $G$ is a group, can there exist an exact covering of $G$ by more than one coset of different sizes?
Erdos274.herzog_schonheim Let $G$ be a group, and let $A = \{a_1G_1, \dots, a_kG_k\}$ be a finite system of left cosets of subgroups $G_1, \dots, G_k$ of $G$
Erdos275.erdos_275 If a finite system of $r$ congruences $\{ a_i\pmod{n_i} : 1\leq i\leq r\}$ (the $n_i$ are not necessarily distinct) covers $2^r$ consecutive integers then it covers all integers
Erdos276.IsLucasSequence We define a Lucas sequence to be a Fibonacci sequence with arbitrary starting points L 0 and L 1
Erdos276.erdos_276 Is there an infinite Lucas sequence
Erdos277.erdos_277 Is it true that, for every $c$, there exists an $n$ such that $\sigma(n)>cn$ but there is no covering system whose moduli all divide $n$?
Erdos279.erdos_279 Let $k\geq 3$
Erdos28.erdos_28 If $A ⊆ \mathbb{N}$ is such that $A + A$ contains all but finitely many integers then $\limsup 1_A ∗ 1_A(n) = \infty$
Erdos280.isCoveredBy The integer m is covered by one of the first k chosen congruence classes
Erdos280.uncoveredCount The number of integers below n k not covered by the first k congruence classes
Erdos281.ResidueChoice Choices of congruence classes $a_i \pmod{n_i}$
Erdos281.avoidAll The integers avoiding all congruences $a_i \pmod{n_i}$
Erdos281.avoidPrefix The integers avoiding the first $k$ congruences $a_i \pmod{n_i}$
Erdos281.erdos_281 Documented declaration: erdos_281
Erdos282.greedyUnitFractionRem Documented declaration: greedyUnitFractionRem
Erdos282.erdos_282 Documented declaration: erdos_282
Erdos282.erdos_282.variants.general More generally, for which pairs $x$ and $A$ does this process terminate?
Erdos282.erdos_282.variants.fibonacci In 1202 Fibonacci observed that this process terminates for any $x$ when $A=\mathbb{N}$
Erdos282.erdos_282.variants.graham Documented declaration: graham
Erdos282.erdos_282.variants.sq Graham has also shown that $x$ is the sum of distinct unit fractions with square denominators if and only if $x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$
Erdos283.Condition Given a polynomial p with rational coefficients, the predicate that if p takes integer values at all integers, the leading coefficient is positive and there exists no
Erdos283.erdos_283 Documented declaration: erdos_283
Erdos283.erdos_283.variants.graham Graham [Gr63] has proved this when $p(x)=x$
Erdos283.erdos_283.variants.graham_alpha Graham also conjectures that this remains true with $1$ replaced by an arbitrary rational $\alpha>0$ (provided $m$ is taken sufficiently large depending on $\alpha$)
Erdos283.erdos_283.variants.cassels Cassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of $p(n_i)$ with distinct $n_i$
Erdos283.erdos_283.variants.burr Burr has proved this if $p(x)=x^k$ with $k\geq 1$ and if we allow $n_i=n_j$
Erdos283.erdos_283.variants.alekseyev Alekseyev [Al19] has proved this when $p(x)=x^2$, for all $m>8542$
Erdos283.erdos_283.variants.van_doorn_linear van Doorn [vD25] has investigated the question of what 'sufficiently large' means for $p(x)=x$
Erdos283.erdos_283.variants.van_doorn_quadratic van Doorn [vD25] has proved the original conjecture for many linear and quadratic polynomials
Erdos285.erdos_285 Documented declaration: erdos_285
Erdos285.erdos_285.variants.lb It is trivial that $f(k)\geq (1 + o(1)) \frac{e}{e - 1}k$
Erdos287.max_gap The maximum gap between consecutive terms of a finite sequence s : Fin k → ℕ, i.e., $\max_{0 \le i < k-1} (s(i+1) - s(i))$
Erdos287.erdos_287 Let $k\geq2$
Erdos287.erdos_287.test.best_possible Documented declaration: best_possible
Erdos287.erdos_287.variants.prime_conjecture For all large $N$, there exists a prime $p \in [N, 2N]$ such that $\frac{p+1}{2}$ is also prime
Erdos288.erdos_288 Is it true that there are only finitely many pairs of intervals $I_1$, $I_2$ such that $$ \sum_{n_1 \in I_1} \frac{1}{n_1} + \sum_{n_2 \in I_2} \frac{1}{n_2} \in \mathbb{N}? $$
Erdos288.erdos_288.variants.i2_card_eq_1 This is still open even if $|I_2| = 1$
Erdos288.erdos_288.variants.k_intervals It is perhaps true with two intervals replaced by any $k$ intervals
Erdos288.erdos_288.variants.exists_k_gt_2 Is it true for any $k > 2$ that only finitely many $k$ intervals satisfy this condition?
Erdos290.harmonicDen The denominator (in lowest terms) of the partial harmonic sum $\sum_{a \leq n \leq b}\frac{1}{n}$
Erdos290.erdos_290 Let $a\geq 1$
Erdos291.L $L_n$ is the least common multiple of $\{1,\ldots,n\}$
Erdos291.a $a_n$ is defined by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}$
Erdos291.erdos_291.parts.i Let $n\geq 1$ and define $L_n$ to be the least common multiple of $\{1,\ldots,n\}$ and $a_n$ by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}$
Erdos291.erdos_291.parts.ii Is it true that $(a_n,L_n)>1$ occurs for infinitely many $n$?
Erdos291.erdos_291.variants.steinerberger_generalization More generally, if the leading digit of $n$ in base $p$ is $p-1$ then $p\mid (a_n,L_n)$
Erdos291.erdos_291.variants.shiu_heuristic_asymptotic This leads to a heuristic prediction (see for example a preprint of Shiu [Sh16]) of $\asymp\frac{x}{\log x}$ for the number of $n\in [1,x]$ such that $(a_n,L_n)=1$
Erdos291.erdos_291.variants.shiu_heuristic_density_zero In particular, there should be infinitely many $n$, but the set of such $n$ should have density zero
Erdos291.erdos_291.variants.wu_yan Documented declaration: wu_yan
Erdos292.A The set $A$ of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$
Erdos292.erdos_292.variants.martin Martin [Ma00] proved that if $B=\mathbb{N}\backslash A$ then $\frac{\lvert B\cap [1,x]\rvert}{x}\asymp \frac{\log\log x}{\log x}$
Erdos292.erdos_292.variants.mul Straus observed that $A$ is closed under multiplication
Erdos292.erdos_292.variants.prime_pow $A$ does not contain any prime power
Erdos292.erdos_292.variants.two_mul van Doorn observed that if $n\in A$ (with $n>1$) then $2n\in A$ also
Erdos295.exists_k Helper lemma: for each $N$, there exists $k$ and $n_1 < ... < n_k$ such that $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$
Erdos295.k Let $k(N)$ denote the smallest $k$ such that there exists $N ≤ n_1 < ⋯ < n_k$ with $\frac 1 {n_1} + ... + \frac 1 {n_k} = 1$
Erdos295.erdos_295 Documented declaration: erdos_295
Erdos295.erdos_295.variants.erdos_straus Erdős and Straus have proved the existence of some constant $c>0$ such that $-c < k(N)-(e-1)N \ll \frac N {\log N}$
Erdos296.recipSum The reciprocal sum of a finite set of natural numbers, as a rational number
Erdos296.HasDisjointUnitDecomps There are k pairwise disjoint subsets of {1, ..., N} with reciprocal sum 1
Erdos298.erdos_298 Does every set $A \subseteq \mathbb{N}$ of positive density contain some finite $S \subset A$ such that $\sum_{n \in S} \frac{1}{n} = 1$?
Erdos298.erdos_298.variants.natural_density The literal natural-density interpretation of Erdős Problem 298 follows from [Bl21]
Erdos299.erdos_299 Is there an infinite sequence $a_1 < a_2 < \dots$ such that $a_{i+1} - a_i = O(1)$ and no finite sum of $\frac{1}{a_i}$ is equal to 1?
Erdos299.erdos_299.variants.density The corresponding question is also false if one replaces sequences such that $a_{i+1} - a_i = O(1)$ with sets of positive density, as follows from [Bl21]
Erdos3.erdos_3 Erdos Problem 3: divergent reciprocal sums and arithmetic progressions
Erdos3.erdos_3.variants.three The case of $3$-term progressions: if $A \subset \mathbb{N}$ has $\sum_{n \in A}\frac 1 n = \infty$, then $A$ contains a non-trivial $3$-term arithmetic progression
Erdos3.r $r_k(N)$ is the largest size of a subset of $\{1, \dots, N\}$ that does not contain a non-trivial $k$-term arithmetic progression
Erdos3.erdos_3.variants.kelley_meka There are $\beta > 0$ and $c > 0$ such that $r_3(N) \ll N \exp(-c (\log N)^{\beta})$
Erdos3.erdos_3.variants.green_tao There is $c > 0$ such that $r_4(N) \ll N (\log N)^{-c}$
Erdos3.erdos_3.variants.gowers For every $k \geq 3$ there is $c_k > 0$ such that $r_k(N) \ll_k N (\log \log N)^{-c_k}$
Erdos3.erdos_3.variants.leng_sah_sawhney For every $k \geq 5$ there is $c_k > 0$ such that $r_k(N) \ll_k N \exp(-(\log \log N)^{c_k})$
Erdos30.h Let $h(N)$ be the maximum size of a Sidon set in $\{1, \dots, N\}$
Erdos30.erdos_30 Is it true that, for every $\varepsilon > 0$, $h(N) = \sqrt N + O_{\varepsilon}(N^\varepsilon)$
Erdos30.erdos_30.variants.O_one A stronger conjecture: is it true that $h(N) = \sqrt N + O(1)$?
Erdos30.erdos_30.variants.erdos_turan Erdős and Turán [ErTu41] proved $h(N) \le \sqrt N + O(N^{1/4})$
Erdos30.erdos_30.variants.lindstrom The proofs of Erdős–Turán [ErTu41] and Lindström [Li69] in fact give, for all $N$, $h(N) \le N^{1/2} + N^{1/4} + 1$
Erdos30.erdos_30.variants.balogh_furedi_roy Balogh, Füredi and Roy [BFR23] proved $h(N) \le N^{1/2} + 0.998 N^{1/4}$ for all sufficiently large $N$
Erdos30.erdos_30.variants.obryant O'Bryant [OB22] proved $h(N) \le N^{1/2} + 0.99703 N^{1/4}$ for all sufficiently large $N$
Erdos30.erdos_30.variants.carter_hunter_obryant Carter, Hunter and O'Bryant [CHO25] proved $h(N) \le N^{1/2} + 0.98183 N^{1/4} + O(1)$
Erdos30.erdos_30.variants.singer Singer's construction [Si38] shows $h(N) \ge (1 - o(1)) N^{1/2}$ for all $N$
Erdos30.erdos_30.variants.isEquivalent_sqrt Combining Singer's lower bound [Si38] with the Erdős–Turán upper bound [ErTu41]: $h(N) \sim N^{1/2}$
Erdos300.A $A(N)$ is the maximal cardinality of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in S}\frac{1}{n}\neq 1$ for all $S\subseteq A$
Erdos302.NoUnitFractionTriple A finite set $A$ of positive integers admits no solution to $\frac{1}{a} = \frac{1}{b} + \frac{1}{c}$ with $a, b, c$ distinct elements of $A$
Erdos302.IsMaxNoTripleCard $f N$ is the size of the largest $A ⊆ \{1, …, N\}$ containing no solution to $\frac{1}{a} = \frac{1}{b} + \frac{1}{c}$ with distinct $a, b, c ∈ A$
Erdos302.erdos_302.parts.i Let $f(N)$ be the size of the largest $A\subseteq \{1,\ldots,N\}$ such that there are no solutions to $$\frac{1}{a}= \frac{1}{b}+\frac{1}{c}$$ with distinct $a,b,c\in A$?
Erdos302.erdos_302.parts.ii In particular, is $f(N)=(\tfrac{1}{2}+o(1))N$?
Erdos302.erdos_302.variants.lower_half One can take either $A$ to be all odd integers in $[1,N]$ or all integers in $[N/2,N]$ to show $f(N)\geq (1/2+o(1))N$
Erdos302.erdos_302.variants.lower_five_eighths Stijn Cambie has observed that $$f(N)\geq (5/8+o(1))N,$$ taking $A$ to be all odd integers $\leq N/4$ and all integers in $[N/2,N]$
Erdos302.erdos_302.variants.upper_nine_tenths Wouter van Doorn has proved [va25] that $$f(N) \leq (9/10+o(1))N.$$
Erdos303.erdos_303 Is it true that in any finite colouring of the integers there exists a monochromatic solution to $\frac 1 a = \frac 1 b + \frac 1 c$ with distinct $a, b, c$?
Erdos304.unitFractionExpressible The set of k for which a / b can be expressed as a sum of k distinct unit fractions
Erdos304.smallestCollection Documented declaration: smallestCollection
Erdos304.smallestCollection_pos in fact (unitFractionExpressible a b).Nonempty should always be true, but we do not prove it for now
Erdos304.smallestCollectionTo Write $$N(b) = max_{1 \leq a < b} N(a, b)$$
Erdos304.erdos_304.variants.upper_1950 In 1950, Erdős [Er50c] proved the upper bound $$N(b) \ll \log b / \log \log b$$
Erdos304.erdos_304.variants.lower_1950 In 1950, Erdős [Er50c] proved the lower bound $$\log \log b \ll N(b)$$
Erdos304.erdos_304.variants.upper_1985 In 1985 Vose [Vo85] proved the upper bound $$N(b) \ll \sqrt{\log b}$$
Erdos304.upper_bound Is it true that $$N(b) \ll \log \log b$$?
Erdos305.D D a b is the minimal value of $n_k$ such that there exist integers $1\leq n_1<\cdots <n_k$ with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$
Erdos305.Dmax $D(b)=\max_{1\leq a<b}D(a,b)$
Erdos305.erdos_305.variants.bleicher_erdos Bleicher and Erdős [BlEr76] have shown that $D(b)\ll b(\log b)^2$
Erdos305.erdos_305.variants.prime_lower_bound If $b=p$ is a prime then $D(p) \gg p\log p$
Erdos305.erdos_305.variants.yokota Yokota [Yo88] proved that $D(b)\ll b(\log b)(\log\log b)^4(\log\log\log b)^2$
Erdos305.erdos_305.variants.liu_sawhney Liu and Sawhney [LiSa24] proved that $D(b)\ll b(\log b)(\log\log b)^3(\log\log\log b)^{O(1)}$
Erdos306.erdos_306 Let $\frac a b\in \mathbb{Q}_{>0}$ with $b$ squarefree
Erdos306.erdos_306.variants.integer_three_primes Every positive integer can be expressed as an Egyptian fraction where each denominator is the product of three distinct primes
Erdos307.erdos_307 Are there two finite set of primes $P$ and $Q$ such that $$ 1 = \left( \sum_{p \in P} \frac{1}{p} \right) \left( \sum_{q \in Q} \frac{1}{q} \right) $$ ?
Erdos307.erdos_307.variants.coprime Instead of asking for sets of primes, ask only that all elements in the sets be relatively coprime
Erdos307.erdos_307.variants.coprime_one_notMem There are no examples known of the weakened coprime version if we insist that $1\not\in P\cup Q$
Erdos308.IsRepresentable k is representable as a sum of distinct unit fractions with denominators from {1, …, N}
Erdos308.representable The set of positive integers representable as a sum of distinct unit fractions with denominators from {1, …, N}
Erdos308.f f N is the smallest positive integer which is not representable as a sum of distinct unit fractions with denominators from {1, …, N}
Erdos308.m m N = ⌊∑_{n ≤ N} 1/n⌋
Erdos308.erdos_308.variants.croot Croot [Cr99] bounded
Erdos308.erdos_308.variants.all_N Documented declaration: all_N
Erdos309.F The number of integers which can be written as the sum of distinct unit fractions with denominators from $\{1,\ldots,N\}$
Erdos310.erdos_310.variants.liu_sawhney Documented declaration: liu_sawhney
Erdos312.erdos_312 Documented declaration: erdos_312
Erdos313.erdos313Solutions This set contains all solutions (m, P) to the Erdős problem 313
Erdos313.erdos_313 Are there infinitely many pairs (m, P) where m ≥ 2 is an integer and P is a set of distinct primes such that the following equation holds
Erdos313.IsPrimaryPseudoperfect An integer n is a primary pseudoperfect number if it is the denominator m in a solution (m, P) to the Erdős 313 problem
Erdos313.erdos_313.variants.primary_pseudoperfect_are_infinite It is conjectured that the set of primary pseudoperfect numbers is infinite
Erdos313.erdos_313.variants.exists_at_least_eight_primary_pseudoperfect There are at least 8 primary pseudoperfect numbers
Erdos314.mMin mMin n is the minimal m such that $\sum_{n\leq k\leq m}\frac{1}{k}\geq 1$; such an m exists for n ≥ 1 since the harmonic series diverges
Erdos314.epsilon $\epsilon(n) = \sum_{n\leq k\leq m}\frac{1}{k}-1$, the overshoot of the minimal block sum reaching $1$
Erdos314.erdos_314 Let $n\geq 1$ and let $m$ be minimal such that $\sum_{n\leq k\leq m}\frac{1}{k}\geq 1$
Erdos315.u Documented declaration: u
Erdos315.c₀ The Vardi constant $c_0=\lim u_n^{1/2^n}=1.264085\cdots$
Erdos315.erdos_315 Documented declaration: erdos_315
Erdos316.erdos_316 Documented declaration: erdos_316
Erdos316.erdos_316.variants.multiset This is not true if $A$ is a multiset, for example $2,3,3,5,5,5,5$
Erdos316.erdos_316.variants.generalized This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of $120$ form a counterexample
Erdos317.erdos_317 Is there some constant
Erdos317.erdos_317.variants.claim2 Is it true that for sufficiently large
Erdos317.claim2_inequality Inequality in erdos_317.variants.claim2 is obvious, the problem is strict inequality
Erdos317.erdos_317.variants.counterexample erdos_317.variants.claim2 fails for small $n$, for example $$\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.$$
Erdos318.P₁ Documented declaration: P₁
Erdos318.erdos_318.variants.squares The set of squares does not have property P₁
Erdos318.erdos_318.variants.contain_single_even For any infinite set A containing exactly one positive even number, A does not have property P₁
Erdos318.erdos_318.parts.i There exists a set A with positive density that does not have property P₁
Erdos318.erdos_318.parts.ii Does the set of squares excluding 1 have property P₁?
Erdos319.erdos_319 What is the size of the largest
Erdos319.erdos_319.variants.isTheta Documented declaration: isTheta
Erdos319.erdos_319.variants.lb Adenwalla has observed that a lower bound (on the maximum size of $A$) of $$ |A| \geq (1 - \frac{1}{e} + o(1))N $$ follows from the main result of Croot [Cr01]
Erdos32.IsAdditiveComplementToPrimes A set $A \subseteq \mathbb{N}$ is an additive complement to the primes if every sufficiently large natural number can be written as $p + a$ for some prime $p$ and $a \in A$
Erdos32.erdos_32.variants.log_squared Erdős proved in [Erd54] that there exists an additive complement $A$ to the primes with $|A \cap \{1, \ldots, N\}| = O((\log N)^2)$
Erdos32.erdos_32.variants.liminf_gt_one Must every additive complement $A$ to the primes satisfy $\liminf_{N \to \infty} \frac{|A \cap \{1, \ldots, N\}|}{\log N} > 1$?
Erdos32.erdos_32 Does there exist a set
Erdos32.erdos_32.variants.log_bound Can the bound $O(\log N)$ be achieved for an additive complement to the primes?
Erdos32.erdos_32.variants.ruzsa Ruzsa proved that any additive complement
Erdos321.R Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$
Erdos321.erdos_321 Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$
Erdos321.erdos_321.variants.isTheta Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$
Erdos321.erdos_321.variants.isBigO Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$
Erdos321.erdos_321.variants.isLittleO Let $R(N)$ be the size of the largest $A\subseteq\{1, ..., N\}$ such that all sums $\sum_{n\in S} \frac{1}{n}$ are distinct for $S\subseteq A$
Erdos321.erdos_321.variants.lower Let $R(N)$ be the maximal such size
Erdos321.erdos_321.variants.upper Let $R(N)$ be the maximal such size
Erdos322.representationCount For k ≥ 3, the number of ordered representations of n as a sum of k many kth powers of positive integers
Erdos322.erdos_322 Let $k\geq 3$ and $A\subset \mathbb{N}$ be the set of $k$th powers
Erdos323.f Let $1\leq m\leq k$ and $f_{k,m}(x)$ denote the number of integers $\leq x$ which are the sum of $m$ many nonnegative $k$th powers
Erdos323.erdos_323.parts.i Is it true that $f_{k,k}(x) \gg_\epsilon x^{1-\epsilon}$ for all $\epsilon>0$?
Erdos323.erdos_323.parts.ii Is it true that if $m < k$ then $f_{k,m}(x) \gg x^{m/k}$ for sufficiently large $x$?
Erdos323.erdos_323.variants.k_eq_2 The case $k=2$ was resolved by Landau, who showed $f_{2,2}(x) \sim \frac{cx}{\sqrt{\log x}}$ for some constant $c>0$
Erdos323.erdos_323.variants.k_gt_2 For $k>2$ it is not known if $f_{k,k}(x)=o(x)$
Erdos324.erdos_324 Does there exist a polynomial $f(x)\in\mathbb{Z}[x]$ such that all the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct?
Erdos324.erdos_324.variants.quintic Probably $f(x) = x^5$ has the property that the sums $f(a)+f(b)$ with $a < b$ nonnegative integers are distinct
Erdos325.IsSumThreePower A predicate for $n$ to be the sum of three $k$th powers
Erdos325.cardIsSumThreePowerBelow The number of integers $\leq x$ which are the sum of three $k$th powers
Erdos325.erdos_325 Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it true that $f_{k, 3}(x) \gg x ^ (3 / k)$?
Erdos325.erdos_325.variants.weaker Writing $f_{k, 3}(x)$ for the number of integers $\leq x$ which are the sum of three $k$th powers, is it even true that $f_{k, 3}(x) \gg_{\epsilon} x ^ (3 / k - \epsilon)$?
Erdos325.erdos_325.variants.wooley For $k = 3$, the best known is due to Wooley [Wo15], who proved $f_{3, 3}(x) \gg x^{0.91709477}$
Erdos326.erdos_326 Does there exist
Erdos326.erdos_326.variants.eq Erdős originally asked this for any basis (not necessarily minimal); such a basis was constructed by Cassels [Ca57]
Erdos328.erdos_328 Suppose $A\subseteq\mathbb{N}$ and $C>0$ is such that $1_A\ast 1_A(n)\leq C$ for all $n\in\mathbb{N}$
Erdos329.sqrtPartialDensity The partial density of a Sidon set A up to N, normalized by dividing by √N instead of N
Erdos329.sidonUpperDensity The upper density of a Sidon set A, normalized by √N
Erdos329.erdos_329 Erdős Problem 329
Erdos329.erdos_329.variants.lower_bound Erdős proved that upper density 1 / 2 can be attained; in particular, there exists a Sidon set whose upper density is at least 1 / 2
Erdos329.erdos_329.variants.kruckeberg_1961 Krückeberg ([Kr61]) exhibited an infinite Sidon set A with sidonUpperDensity A = 1 / Real.sqrt 2, improving Erdős’ earlier 1 / 2 lower bound
Erdos329.erdos_329.variants.turan_1941 Erdős and Turán [ErTu41] proved the upper bound of 1
Erdos329.erdos_329.variants.converse_implication The converse: if the maximum density is 1, then any finite Sidon set can be embedded in a perfect difference set modulo $n > 0$
Erdos329.exists_sidon_pos_density It is possible to construct a Sidon set with positive density
Erdos33.AdditiveBasisCondition Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0
Erdos33.erdos_33 Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0
Erdos33.erdos_33.variants.one_mem_lowerBounds Erdos observed that this value is finite and > 1
Erdos33.erdos_33.variants.vanDoorn The smallest possible value of lim sup n → ∞ |A ∩ {1, …, N}| / N^(1/2) is at most 2φ^(5/2) ≈ 6.66, with φ equal to the golden ratio
Erdos330.Rep Rep A m h means m is a sum of exactly h elements of A, which is m ∈ h • A
Erdos330.UnrepWithout Integers not representable as a sum of exactly h elements of A while avoiding n
Erdos330.MinAsymptoticAddBasisOfOrder An asymptotic additive basis of order h is minimal when one cannot obtain an asymptotic additive basis by removing any element from it
Erdos331.erdos_331 Documented declaration: erdos_331
Erdos331.erdos_331.variants.ruzsa Documented declaration: ruzsa
Erdos332.D_A The set of numbers $D(A)$ which occur infinitely often as $a_1 - a_2$ with $a_1, a_2 \in A$
Erdos332.HasBoundedGaps A set $S \subseteq \mathbb{Z}$ has bounded gaps if it is syndetic, meaning there is a uniform bound $M$ such that every interval of length $M$ contains an element of $S$
Erdos332.erdos_332 Let $A\subseteq \mathbb{N}$ and $D(A)$ be the set of those numbers which occur infinitely often as $a_1 - a_2$ with $a_1, a_2\in A$
Erdos333.erdos_333 Let $A\subseteq \mathbb{N}$ be a set of density zero
Erdos337.erdos_337 Let $A\subseteq \mathbb{N}$ be an additive basis (of any finite order) such that $\lvert A\cap \{1,\ldots,N\}\rvert=o(N)$
Erdos337.erdos_337.variants.h_fold This was generalised (to the replacement of $A+A$ by the $h$-fold sumset $hA$ for any $h\geq 2$) by Ruzsa and Turjányi [RT85]
Erdos34.consecutiveSums The set of consecutive sums of a permutation p of {1,…,n}, where position k holds the value p k + 1: the sums ∑_{u ≤ k ≤ v} (p k + 1)
Erdos340.erdos_340 Documented declaration: erdos_340
Erdos340.erdos_340.variants.isTheta Documented declaration: isTheta
Erdos340.erdos_340.variants.third It is trivial that this sequence grows at least like $\gg N^{1/3}$
Erdos340.erdos_340.variants.sub_hasPosDensity Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this has positive density
Erdos340.erdos_340.variants._22_mem_sub Erdős and Graham [ErGr80] also asked about the difference set $A - A$ and whether this contains $22$, which it does
Erdos340.erdos_340.variants._33_mem_sub The smallest integer which is unknown to be in $A - A$ is $33$
Erdos340.erdos_340.variants.cofinite_sub It may be true that all or almost all integers are in $A - A$
Erdos340.erdos_340.variants.co_density_zero_sub It may be true that all or almost all integers are in $A - A$
Erdos342.UniqueUlamSum UniqueUlamSum a n m means that $m$ has a unique representation as $a(i) + a(j)$ with $i < j < n$
Erdos342.IsUlamSequence Documented declaration: IsUlamSequence
Erdos342.erdos_342.test.a0 $a(0) = 1$ by definition
Erdos342.erdos_342.test.a1 $a(1) = 2$ by definition
Erdos342.erdos_342.test.a2 $a(2) = 3$: the only pair $(i,j)$ with $i < j < 2$ is $(0,1)$, giving $1 + 2 = 3$
Erdos342.erdos_342.test.a3 $a(3) = 4$: among sums $> 3$ with a unique representation from $\{1,2,3\}$, the smallest is $4 = 1 + 3$
Erdos342.erdos_342.parts.i Do infinitely many pairs $(a, a+2)$ occur in Ulam's sequence?
Erdos342.erdos_342.parts.ii Does Ulam's sequence eventually have periodic differences?
Erdos342.erdos_342.parts.iii Part (iii), is the density of the sequence 0?
Erdos346.erdos_346 Is it true that for every lacunary, strongly complete sequence A that is not complete whenever infinitely many terms are removed from it, lim A (n + 1) / A n = (1 + √5) / 2?
Erdos346.f We define a sequence f by the formula f n = n.fib - (- 1) ^ n
Erdos346.erdos_346.variants.f_isLacunary The sequence f is lacunary
Erdos346.erdos_346.variants.f_isAddStronglyCompleteNatSeq The sequence f is strongly complete, and this is proved in [Gr64d]
Erdos346.erdos_346.variants.f_add_two The recurrence f (m + 2) = f (m + 1) + f m - (-1) ^ m for m ≥ 1, written without subtraction
Erdos346.erdos_346.variants.sum_range_f f 0 + ⋯ + f (m - 1) = f (m + 1) - [m even] for m ≥ 1; this is [Gr64d, Eq. (1)]
Erdos346.erdos_346.variants.f_strictMono f is strictly increasing from index 4 on
Erdos346.erdos_346.variants.f_not_isAddComplete The sequence f is not complete whenever infinitely many terms are removed from it, and this is proved in [Gr64d]
Erdos346.erdos_346.variants.gt_goldenRatio_not_IsAddComplete Erdős and Graham [ErGr80] remark that it is easy to see that if A (n + 1) / A n > (1 + √5) / 2 then the second property is automatically satisfied
Erdos346.erdos_346.variants.example Documented declaration: example
Erdos347.erdos_347 Is there a sequence
Erdos348.erdos_348 For what values of $0 \leq m < n$ is there a complete sequence $A = \{a_1 \leq a_2 \leq \cdots\}$ of integers such that 1
Erdos349.IsGoodPair This defines the core property of the problem: For what values of $t,\alpha \in (0,\infty)$ is the sequence $\lfloor t\alpha^n\rfloor$ complete?
Erdos349.erdos_349 Documented declaration: erdos_349
Erdos349.complete_for_alpha_in_Ioo_one_to_goldenRatio It seems likely that the sequence is complete for all for all $t>0$ and all $1 < \alpha < \frac{1+\sqrt{5}}{2}$
Erdos349.exists_t_for_k_disjoint_segments Documented declaration: exists_t_for_k_disjoint_segments
Erdos349.erdos_349.variants.floor_3_halves_odd Is it true that the terms of the sequence $\lfloor (3/2)^n\rfloor$ are odd infinitely often and even infinitely often?
Erdos349.erdos_349.variants.floor_3_halves_even Is it true that the terms of the sequence $\lfloor (3/2)^n\rfloor$ are even infinitely often?
Erdos35.erdos_35.variants.plunnecke Documented declaration: plunnecke
Erdos350.DistinctSubsetSums The predicate that all (finite) subsets of A have distinct sums
Erdos350.DecidableDistinctSubsetSums The predicate that all (finite) subsets of A have distinct sums, decidable version
Erdos350.DistinctSubsetSums_iff_DecidableDistinctSubsetSums Small sanity check: the two predicates are saying the same thing
Erdos350.erdos_350.variants.strengthening If A ⊂ ℕ is a finite set of integers all of whose subset sums are distinct then ∑ n ∈ A, 1/n^s < 1/(1 - 2^(-s)), for any s > 0
Erdos351.imageSet The set of rational numbers of the form P(n) + 1 / n where n is a natural number and P is a polynomial with rational coefficients
Erdos351.IsStronglyComplete The predicate that a set A is strongly complete, i.e. that for every finite set B, every sufficiently large integer is a sum of elements of the set A \ B
Erdos351.HasCompleteImage The predicate that the rational polynomial P has a complete image
Erdos351.erdos_351 Let $p(x) \in \mathbb{Q}[x]$ be a non-constant rational polynomial with positive leading coefficient
Erdos351.erdos_351.variants.X Let $p(x) = x \in \mathbb{Q}[x]$
Erdos351.erdos_351.variants.X_sq Let $p(x) = x ^ 2 \in \mathbb{Q}[x]$
Erdos352.erdos_352 Is there some $c > 0$ such that every measurable $A \subseteq \mathbb{R}^2$ of measure $\geq c$ contains the vertices of a triangle of area 1?
Erdos353.erdos_353 Let $A\subseteq \mathbb{R}^2$ be a measurable set with infinite measure
Erdos353.erdos_353.variants.isosceles_triangle Every measurable $A\subseteq \mathbb{R}^2$ with infinite measure contains the vertices of an isosceles triangle of area $1$
Erdos353.erdos_353.variants.right_angled_triangle Every measurable $A\subseteq \mathbb{R}^2$ with infinite measure contains the vertices of a right-angled triangle of area $1$
Erdos353.erdos_353.variants.cyclic_quadrilateral Every measurable $A\subseteq \mathbb{R}^2$ with infinite measure contains the vertices of a cyclic quadrilateral of area $1$
Erdos353.erdos_353.variants.congruent_sides Documented declaration: congruent_sides
Erdos354.FloorMultiples The sequence ⌊a⌋, ⌊γ * a⌋, ⌊γ ^ 2 * a⌋, ..., ⌊γ ^ i * a⌋, ...
Erdos354.FloorMultiples.interleave The sequence `⌊a⌋, ⌊b⌋, ⌊γ * a⌋, ⌊γ * b⌋,
Erdos354.erdos_354.parts.i Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational
Erdos354.erdos_354.parts.ii Let $\alpha,\beta\in \mathbb{R}_{>0}$ such that $\alpha/\beta$ is irrational
Erdos355.erdos_355 Is there a lacunary sequence
Erdos357.f Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.parts.i Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.parts.ii.bigO_version Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.parts.ii.bigO_version_symm Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.parts.ii.bigTheta_version Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.parts.ii.littleO_version Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.parts.ii.littleO_version_symm Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.weisenberg Let $f(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 < \dotsc < a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.infinite_set_lower_density Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct
Erdos357.erdos_357.variants.infinite_set_density Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct
Erdos357.erdos_357.variants.infinite_set_sum Suppose $A$ is an infinite set such that all finite sums of consecutive terms of $A$ are distinct
Erdos357.g Let $g(n)$ be the maximal $k$ such that there exist integers $1 \le a_1, \dotsc, a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.hegyvari Let $g(n)$ be the maximal $k$ such that there exist integers $1 \le a_1, \dotsc, a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.h Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.monotone.parts.i Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.monotone.parts.ii.bigO_version Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.monotone.parts.ii.bigO_version_symm Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.monotone.parts.ii.bigTheta_version Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.monotone.parts.ii.littleO_version Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos357.erdos_357.variants.monotone.parts.ii.littleO_version_symm Let $h(n)$ be the maximal $k$ such that there exist integers $1 \le a_1 \leq \dotsc \leq a_k \le n$ such that all sums of the shape $\sum_{u \le i \le v} a_i$ are distinct
Erdos358.intervalRepresentations Let $a$ be an infinite sequence of integers
Erdos358.f Let $a$ be an infinite sequence of integers
Erdos358.intervalRepresentationsNonTrivial Let $a$ be an infinite sequence of integers
Erdos358.g Let $a$ be an infinite sequence of integers
Erdos358.f_id When $A_n = n$, the function $f$ defined above counts the number of odd divisors of $n$
Erdos358.erdos_358.parts.i Let $A=\{a_1 < \cdots\}$ be an infinite sequence of integers
Erdos358.erdos_358.parts.ii Let $A=\{a_1 < \cdots\}$ be an infinite sequence of integers
Erdos358.erdos_358.variants.prime_set When $A =\{a_1 < \cdots\}$ corresponds to the set of primes, it is conjectured that the $\limsup$ of the number of representations $$n=\sum_{u\leq i\leq v}a_i$$ is infinite
Erdos358.erdos_358.variants.prime_set_density_representation Documented declaration: prime_set_density_representation
Erdos358.one_le_g_of_two_le_f Documented declaration: one_le_g_of_two_le_f
Erdos358.erdos_358.variants.one_le In [ErGr80] Erdős and Graham further asked whether there is an $A$ with $f(n)\geq 1$ for all large $n$
Erdos359.IsGoodFor The predicate that A is monotone, A 0 = n and for all j, A (j + 1) is the smallest natural number that cannot be written as a sum of consecutive terms of A 0, ..., A j
Erdos359.erdos_359.parts.i Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s
Erdos359.erdos_359.parts.ii Let $a_1< a_2 < ⋯ $ be an infinite sequence of integers such that $a_1=1$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s
Erdos359.erdos_359.variants.isGoodFor_1_low_values Suppose monotone sequence
Erdos359.erdos_359.variants.isGoodFor_1_asymptotic Suppose monotone sequence
Erdos36.Overlap The number of solutions to the equation $a - b = k$, for $a \in A$ and $b \in B$
Erdos36.MaxOverlap The maximum overlap for a given pair of sets $A$ and $B$, taken over all possible integer differences $k$
Erdos36.M Let $A$ and $B$ be two complementary subsets, a splitting of the numbers $\{1, 2, \dots, 2n\}$, such that both have the same cardinality $n$
Erdos36.one_le_overlap A small API lemma: every pair (a, b) ∈ A × B contributes to Overlap A B (a - b)
Erdos36.overlap_le_card_left For a fixed difference k, the first coordinate determines the second, so an overlap is never larger than either side
Erdos36.bddAbove_range_overlap The range of Overlap A B is bounded, so MaxOverlap is a genuine supremum rather than the junk value sSup returns on an unbounded set
Erdos36.overlap_eq_zero_of_notMem_sub An overlap can only be nonzero for a difference that is actually realised, which confines the search for MaxOverlap to a finite set
Erdos36.maxOverlap_eq_sup MaxOverlap as a supremum over a finite set of differences
Erdos36.M_eq_image The pairs M n ranges over are exactly the n-element subsets of {1, …, 2n}, each paired with its complement
Erdos36.maxOverlapC A computable stand-in for MaxOverlap
Erdos36.parts The n-element subsets of {1, …, 2n}
Erdos36.M_eq_min' M n as the minimum of a Finset of naturals, which is what makes it evaluable
Erdos36.maxOverlap_singleton_one_two MaxOverlap {1} {2} = 1: the only nonzero overlap is at k = -1
Erdos36.M_one This example calculates the value of $M 1$
Erdos36.M_two For $n = 2$, the set is $\{1, 2, 3, 4\}$
Erdos36.M_three For n = 3 the best splitting of {1, …, 6} has maximum overlap 2
Erdos36.M_four For n = 4 the best splitting of {1, …, 8} still has maximum overlap 2
Erdos36.M_five For n = 5 the best splitting of {1, …, 10} has maximum overlap 3
Erdos36.MinOverlapQuotient The quotient of the minimum maximum overlap $M(N)$ by $N$
Erdos36.minimum_overlap.variants.lower.erdos_1955 A lower bound of $\frac 1 4$
Erdos36.minimum_overlap.variants.lower.scherk_1955 A lower bound of $1 - frac{1}{\sqrt 2}$
Erdos36.minimum_overlap.variants.lower.swierczkowski_1958 A lower bound of $\frac{4 - \sqrt{6}}{5}$
Erdos36.minimum_overlap.variants.lower.haugland_1996 A lower bound of $\sqrt{4 - \sqrt{15}}$
Erdos36.minimum_overlap.variants.lower.white_2022 A lower bound of $0.379005$: the minimum overlap constant is at least $0.379005$
Erdos36.minimum_overlap.variants.upper.erdos_1955 The example (with $N$ even), $A = \{\frac N 2 + 1, \dots, \frac{3N}{2}\}$ shows an upper bound of $\frac 1 2$
Erdos36.minimum_overlap.variants.upper.MRS_1956 An upper bound of $\frac 2 5$
Erdos36.minimum_overlap.variants.upper.haugland_1996 An upper bound of $0.38200298812318988$
Erdos36.minimum_overlap.variants.upper.haugland_2022 An upper bound of $0.3809268534330870$
Erdos36.erdos_36.variants.lower Find a better lower bound!
Erdos36.erdos_36.variants.upper Find a better upper bound!
Erdos36.erdos_36.variants.exists The limit of MinOverlapQuotient exists and it is less than $0.385694$
Erdos36.erdos_36 Find the value of the limit of MinOverlapQuotient!
Erdos361.AvoidsSubsetSum AvoidsSubsetSum n A means that no subset $B \subseteq A$ has sum equal to $n$
Erdos361.maxSubsetSumAvoidingCard The largest cardinality of a set $A \subseteq \{1, \ldots, N\}$ such that no subset of $A$ sums to $n$
Erdos361.subsetSumAvoidanceNumber For fixed $c$ and $n$, this is the size of the largest set $A \subseteq \{1, \ldots, \lfloor cn \rfloor\}$ such that no subset of $A$ sums to $n$
Erdos361.maxSubsetSumAvoidingCard_three_four For target $4$ and universe $\{1, 2, 3\}$, the maximum is $2$: the full set is invalid because its subset $\{1, 3\}$ sums to $4$
Erdos361.erdos_361 Let $c > 0$ and $n$ be some large integer
Erdos363.IsInterval A finite set of naturals is an interval of naturals
Erdos363.IsValidCollection A collection of intervals as in Erdős Problem 363: disjoint intervals of positive integers, each of size at least 4, whose product is a square
Erdos364.erdos_364 There is no consecutive triple of powerful numbers
Erdos364.erdos_364.variants.strong Erdős [Er76d] conjectured a stronger statement: if $n_k$ is the $k$th powerful number, then $n_{k+2} - n_k > n_k^c$ for some constant $c > 0$
Erdos364.erdos_364.variants.weak There is no quadruple of powerful numbers, since at least one of the four numbers must be $2 \pmod{4}$, which cannot be powerful (since $2$ divides it, but $2^2$ does not)
Erdos365.golomb_pair Golomb's consecutive powerful pair $12167=23^3$ and $12168=2^3 3^2 13^2$
Erdos365.erdos_365.parts.i Do all pairs of consecutive powerful numbers $n$ and $n+1$ come from solutions to Pell equations?
Erdos365.erdos_365.parts.ii Is the number of positive $n\leq x$ such that $n$ and $n+1$ are both powerful bounded by $(\log x)^{O(1)}$?
Erdos366.erdos_366 Are there any $2$-full $n$ such that $n+1$ is $3$-full?
Erdos366.exists_three_full_then_two_full Note that $8$ is $3$-full and $9$ is 2-full
Erdos366.erdos_366.variants.three_two Erdős and Graham asked whether $(8, 9)$ is the only pair of consecutive integers $n$, $n+1$ with $n$ $3$-full and $n+1$ $2$-full
Erdos366.erdos_366.variants.weaker Are there any consecutive pairs of $3$-full integers?
Erdos367.B B r n is the $r$-full part of $n$: the product of prime powers $p^a \| n$ with $a \geq r$
Erdos367.erdos_367.parts.i Let $B_2(n)$ be the $2$-full part of $n$ (that is, $B_2(n)=n/n'$ where $n'$ is the product of all primes that divide $n$ exactly once)
Erdos367.erdos_367.parts.ii Or perhaps even $\prod_{n \leq m < n+k} B_2(m) \ll_k n^2$?
Erdos367.erdos_367.variants.k_le_two van Doorn notes in the comments that for $k \leq 2$ we trivially have $\prod_{n \leq m < n+k} B_2(m) \ll n^2$
Erdos367.erdos_367.variants.k_ge_three_lower But this fails for all $k \geq 3$, and in fact $\prod_{n \leq m < n+3} B_2(m) \gg n^2 \log n$ infinitely often
Erdos367.erdos_367.variants.higher_full_parts Documented declaration: higher_full_parts
Erdos368.F The largest prime factor of $n(n+1)$
Erdos368.erdos_368.lower_bound How large is the largest prime factor of $n(n+1)$?
Erdos368.erdos_368.upper_bound Erdős [Er76d] conjectured that, for every $\epsilon>0$, there are infinitely many $n$ such that $F(n)<(\log n)^{2+\epsilon}$
Erdos368.erdos_368.variants.tendsto Pólya [Po18] proved that $F(n)\to\infty$ as $n\to\infty$
Erdos368.erdos_368.variants.subpower For every $\delta>0$, there are infinitely many $n$ with $F(n)\leq n^\delta$
Erdos369.erdos_369 Let $\epsilon>0$ and $k\geq 2$
Erdos371.erdos_371 Let $P(n)$ denote the largest prime factor of $n$
Erdos372.erdos_372 Let $P(n)$ denote the largest prime factor of $n$
Erdos373.S Let S be the set of non-trivial solutions to the equation `n!
Erdos373.erdos_373 Show that the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, has only finitely many solutions
Erdos373.erdos_373.variants.of_limit Documented declaration: of_limit
Erdos373.erdos_373.variants.of_lower_bound Documented declaration: of_lower_bound
Erdos373.erdos_373.variants.maximal_solution Hickerson conjectured the largest solution the equation n!=a_1!a_2!···a_k!, with n−1 > a_1 ≥ a_2 ≥ ··· ≥ a_k, is 16!=14!5!2!
Erdos373.erdos_373.variants.suranyi Surányi was the first to conjecture that the only non-trivial solution to a!b!=n! is 6!7!=10!
Erdos375.Erdos375Prop This is a proposition saying that for any n ≥ 1 and any k, if n + 1, ..., n + k are all composite, then there are distinct primes `p₁,
Erdos375.erdos_375 Is Erdos375Prop true?
Erdos375.erdos_375.variants.bounded_gap If Erdos375Prop is true, then (n + 1).nth Prime - n.nth Prime < (n.nth Prime) ^ (1 / 2 - c) for some c > 0
Erdos375.erdos_375.variants.legendre In particular, if Erdos375Prop is true, then Legendre's conjecture is asymptotically true
Erdos375.erdos_375.variants.le_two It is easy to see that for any n ≥ 1 and k ≤ 2, if n + 1, ..., n + k are all composite, then there are distinct primes `p₁,
Erdos375.erdos_375.variants.log There exists a constant c > 0 such that for all n, if k < c * (log n / (log (log n))) ^ 3 → (∀ i < k, ¬ (n + i + 1).Prime), then there are distinct primes `p₁,
Erdos376.erdos_376 Are there infinitely many $n$ such that ${2n\choose n}$ is coprime to $105$?
Erdos376.erdos_376.variants.prime Erdős, Graham, Ruzsa, and Straus [EGRS75] have shown that, for any two odd primes $p$ and $q$, there are infinite many $n$ such that ${2n\choose n}$ is coprime to $pq$
Erdos377.sumInvPrimesNotDvdCentralBinom The sum of the inverses of all primes smaller than $n$, which don't divide the central binom coefficient
Erdos377.erdos_377 Is there some absolute constant $C > 0$ such that $$ \sum_{p \leq n} 1_{p\nmid {2n \choose n}}\frac{1}{p} \leq C $$ for all $n$?
Erdos377.erdos_377.variants.limit.i Documented declaration: i
Erdos377.erdos_377.variants.limit.ii Documented declaration: ii
Erdos377.erdos_377.variants.ae Documented declaration: ae
Erdos377.erdos_377.variants.ub Documented declaration: ub
Erdos382.LargestPrimeRepeated Documented declaration: LargestPrimeRepeated
Erdos382.largestPrimeRepeated_48_50 The pair $(48, 50)$ satisfies the condition: $48 \cdot 49 \cdot 50 = 2^5 \cdot 3 \cdot 5^2 \cdot 7^2$, and the largest prime $7$ appears with exponent $2$
Erdos382.lt_two_mul_of_largestPrimeRepeated If the condition holds, then $v < 2u$
Erdos382.erdos_382.parts.i Let $u \leq v$ be such that the largest prime dividing $\prod_{u \leq m \leq v} m$ appears with exponent at least $2$
Erdos382.erdos_382.parts.ii Let $u \leq v$ be such that the largest prime dividing $\prod_{u \leq m \leq v} m$ appears with exponent at least $2$
Erdos382.erdos_382.variants.ramachandra Erdős and Graham [ErGr80] report that it follows from results of Ramachandra that $v - u \leq v^{1/2+o(1)}$ for every pair $u \leq v$ as in Erdos382.erdos_382.parts.i
Erdos383.erdos_383 Is it true that for every $k$ there are infinitely many primes $p$ such that the largest prime divisor of $$ \prod_{i = 0}^k (p ^ 2 + i) $$ is $p$?
Erdos384.erdos_384 If $1k^2$ the binomial coefficient $\binom{n}{k}$ is divisible by a prime $p<n/k$
Erdos385.F Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$
Erdos385.trivial_ub Note that trivially $F(n) \leq n + \sqrt{n}$
Erdos385.erdos_385.parts.i Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$
Erdos385.erdos_385.parts.ii Let $F(n) := \max\{m + p(m) \mid \textrm{$m < n$ composite}\}\}$ where $p(m)$ is the least prime divisor of $m$
Erdos385.erdos_385.variants.lb A question of Erdős, Eggleton, and Selfridge, who write that in fact it is possible that this quantity is always at least $n+(1-o(1))\sqrt{n}$
Erdos386.erdos_386 Let $2 \le k \le n - 2$
Erdos386.erdos_386.variants.forall For all $2 \le k \le n - 2$, can $\binom{n}{k}$ be the product of consecutive primes infinitely often?
Erdos386.erdos_386.variants.two Can $\binom{n}{2}$ be the product of consecutive primes infinitely often?
Erdos387.erdos_387 Is there an absolute constant $c > 0$ such that, for all $1 \leq k < n$, the binomial coefficient $\binom{n}{k}$ has a divisor in $(cn, n]$?
Erdos387.erdos_387.variants.schinzel The following is Schinzel's conjecture, which appears in [Gu04]
Erdos387.erdos_387.variants.easy It is easy to see that n.choose k has a divisor in [n / k, n]
Erdos387.erdos_387.variants.guy Is it true for any $c < 1$ and all $n$ sufficiently large, for all $1 \leq k < n$, $\binom{n}{k}$ has a divisor in $(cn, n]$?
Erdos388.erdos_388 Are there only finitely many solutions of $$\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)$$ with $k_1,k_2>3$ and $m_1+k_1\leq m_2$?
Erdos388.erdos_388.variants.classification Can one classify all solutions of $$\prod_{1\leq i\leq k_1}(m_1+i)=\prod_{1\leq j\leq k_2}(m_2+j)$$ with $k_1,k_2>3$ and $m_1+k_1\leq m_2$?
Erdos388.erdos_388.variants.general More generally, if $k_1>2$ then for fixed $a$ and $b$ the equation $$a\prod_{1\leq i\leq k_1}(m_1+i)=b\prod_{1\leq j\leq k_2}(m_2+j)$$ should have only finitely many solutions
Erdos389.erdos_389 Is it true that for every $n \geq 1$ there is a $k$ such that $$ n(n + 1) \cdots (n + k - 1) \mid (n + k) \cdots (n + 2k - 1)? $$
Erdos389.erdos_389.variants.mehta_four Bhavik Mehta has computed the minimal such $k$ for $1 \leq n \leq 18$
Erdos39.erdos_39 Is there an infinite Sidon set $A\subset \mathbb{N}$ such that $\lvert A\cap \{1\ldots,N\}\rvert \gg_\epsilon N^{1/2-\epsilon}$ for all $\varepsilon > 0$?
Erdos390.f Let f n be the smallest integer for which n! can be represented as the product of distinct integers greater than n, the largest of which is f n
Erdos390.erdos_390.variants.theta f n - 2 * n = θ (n / log n)
Erdos390.erdos_390 Does there exists a constant c such that f n - 2 * n ~ c * (n / log n)?
Erdos391.t t n is the largest t such that n! is a product of n factors all at least t, i.e. the maximal
Erdos392.erdos_392 Let $A(n)$ denote the least value of $t$ such that $$ n! = a_1 \cdots a_t $$ with $a_1 \leq \cdots \leq a_t\leq n^2$
Erdos392.erdos_392.variants.lower If we change the condition to $a_t \leq n$ it can be shown that $$ A(n) = n - \frac{n}{\log n} + o\left(\frac{n}{\log n}\right) $$
Erdos392.erdos_392.variants.implication Documented declaration: implication
Erdos394.t Let $t_k(n)$ denote the least $m$ such that $n\mid m(m+1)(m+2)\cdots (m+k-1).$
Erdos394.t_eq_of t k n = v when v works and nothing positive below it does
Erdos394.t_one The least positive multiple of n is n, so t 1 n = n
Erdos394.erdos_394.parts.i Is it true that $\sum_{n\leq x}t_2(n)\ll \frac{x^2}{(\log x)^c}$ for some $c>0$?
Erdos394.erdos_394.parts.ii Is it true that, for $k\geq 2$, $\sum_{n\leq x}t_{k+1}(n) =o\left(\sum_{n\leq x}t_k(n)\right)?$
Erdos394.erdos_394.variants.hall_bound In [ErGr80] they mention a conjecture of Erdős that the sum is $o(x^2)$
Erdos394.erdos_394.variants.hall_conjecture Erdős and Hall conjecture that the sum is $o(x^2/(\log x)^c)$ for any $c<\log 2$
Erdos394.erdos_394.variants.lower_bound Since $t_2(p)=p-1$ for prime $p$ it is trivial that $\sum_{n\leq x}t_2(n)\gg \frac{x^2}{\log x}$
Erdos394.erdos_394.variants.factorial_gap_conjecture They ask about the behaviour of $t_{n-3}(n!)$ and also ask whether, for infinitely many $n$, $t_k(n!)< t_{k-1}(n!)-1$ for all $1\leq k < n$
Erdos394.erdos_394.variants.factorial_gap_10 They proved (with Selfridge) that this holds for $n=10$
Erdos395.signedSumCount The number of sign patterns $\epsilon \in \{-1,1\}^n$ with $\lvert \epsilon_1z_1+\cdots+\epsilon_nz_n\rvert \leq r$
Erdos395.erdos_395.variants.one Documented declaration: one
Erdos395.erdos_395.variants.sharp The bound of $1/n$ in erdős_395 is the best possible, as shown by taking $z_k=1$ for $1\leq k\leq n/2$ and $z_k=i$ otherwise
Erdos396.erdos_396 Is it true that for every $k$ there exists $n$ such that $$\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$$
Erdos398.erdos_398 Brocard's Problem Does $n! + 1 = m^2$ have integer solutions other than $n = 4, 5, 7$?
Erdos399.erdos_399 Is it true that there are no solutions to `n!
Erdos399.erdos_399.variants.erdos_oblath Erdős and Obláth [ErOb37] proved this is true when $(x,y)=1$ and $k\neq 4$
Erdos399.erdos_399.variants.pollack_shapiro Pollack and Shapiro [PoSh73] proved there are no solutions to $n!=x^4-1$
Erdos399.erdos_399.variants.cambie Cambie has also observed that considerations modulo $8$ rule out any solutions to $n!=x^4+y^4$ with $(x,y)=1$ and $xy>1$
Erdos399.erdos_399.variants.sum_two_squares Documented declaration: sum_two_squares
Erdos4.erdos_4 Is it true that, for any $C > 0$, there infinitely many $n$ such that: $$ p_{n + 1} - p_n > C \frac{\log\log n\log\log\log\log n}{(\log\log\log n) ^ 2}\log n $$
Erdos4.erdos_4.variants.rankin Rankin's theorem: there exists a positive constant $C$ such that Erdos4For C holds
Erdos40.Erdos40For The predicate for a function $g\colon\mathbb{N} → \mathbb{R})$ that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$
Erdos40.erdos_40 For what functions $g(N) → \infty$ is it true that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$?
Erdos40.erdos_40.variants.weaker Is there any function $g(N) → \infty$ such that $$\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$$ implies $\limsup 1_A\ast 1_A(n)=\infty$?
Erdos40.erdos_40.variants.implies_erdos_28 Establishing the property in Erdős Problem 40 for any one function $g(N) → \infty$ implies the Erdős-Turán conjecture, see Erdős Problem 28
Erdos400.g For any $k\geq 2$ let $g_k(n)$ denote the maximum value of $(a_1+\cdots+a_k)-n$ where $a_1,\ldots,a_k$ are integers such that $a_1!\cdots a_k! \mid n!$
Erdos400.erdos_400.parts.i Can one show that $\sum_{n\leq x}g_k(n) \sim c_k x\log x$ for some constant $c_k$?
Erdos400.erdos_400.parts.ii Is it true that there is a constant $c_k$ such that for almost all $n < x$ we have $g_k(n)=c_k\log x+o(\log x)$?
Erdos400.erdos_400.variants.upper_bound Erdős and Graham write that it is easy to show that $g_k(n) \ll_k \log n$ always, but the best possible constant is unknown
Erdos400.erdos_400.variants.g_pos For $k \ge 2$, $g_k(n) > 0$
Erdos401.erdos_401 Is there some function
Erdos402.erdos_402 Prove that, for any finite set $A\subset\mathbb{N}$, there exist $a, b\in A$ such that $$ \gcd(a, b)\leq a/|A|. $$
Erdos402.erdos_402.variants.equality A conjecture of Graham [Gr70], who also conjectured that (assuming
Erdos402.erdos_402.variants.szegedy_zaharescu_weak Proved for all sufficiently large sets (including the sharper version which characterises the case of equality) independently by Szegedy [Sz86] and Zaharescu [Za87]
Erdos403.erdos_403 Does the equation $$2^m=a_1!+\cdots+a_k!$$ with $a_1 0$
Erdos405.erdos_405 Let $p$ be an odd prime
Erdos405.erdos_405.variants.nonodd_p Erdős and Graham [ErGr80] ask this allowing $p=2$, but this is presumably an oversight, since clearly there are infinitely many solutions when $p=2$
Erdos405.erdos_405.variants.yu_liu Yu and Liu [YuLi96] showed that the only solutions to $(p-1)! + a^(p-1) = p^k$ for an odd prime p are: $$ 2! + 1^2 = 3\\ 2! + 5^2 = 3^3\\ 4! + 1^4 = 5^2 $$
Erdos406.erdos_406 Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?
Erdos406.erdos_406.variants.one_two If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$
Erdos407.w w n counts the ordered quadruples $(a,b,c,d)$ of nonnegative integers with $n=2^a+3^b+2^c3^d$
Erdos409.erdos_409.parts.i How many iterations of $n\mapsto\phi(n) + 1$ are needed before a prime is reached?
Erdos409.erdos_409.variants.termination If $n > 0$, then the iteration $n\mapsto\phi(n) + 1$ necessarily reaches a prime
Erdos409.erdos_409.parts.i.isTheta Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached
Erdos409.erdos_409.parts.i.isBigO Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached
Erdos409.erdos_409.parts.i.isLittleO Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached
Erdos409.erdos_409.parts.ii Can infinitely many $n$ reach the same prime under the iteration $n\mapsto\phi(n) + 1$?
Erdos409.erdos_409.parts.iii What is the density of $n$ which reach any fixed prime under the iteration $n\mapsto\phi(n) + 1$?
Erdos409.erdos_409.variants.sigma If the iteration $n\mapsto\sigma(n) - 1$ starting at $n > 1$ reaches a prime, how many iterations are needed before a prime is reached?
Erdos409.erdos_409.variants.sigma_termination If $n > 1$ then the iteration $n\mapsto\sigma(n) - 1$ necessarily reaches a prime
Erdos409.erdos_409.variants.sigma_isTheta Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached
Erdos409.erdos_409.variants.sigma_isBigO Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached
Erdos409.erdos_409.variants.sigma_isLittleO Let $c(n)$ be the minimum number of iterations of $n\mapsto\sigma(n) - 1$ before a prime is reached
Erdos409.erdos_409.variants.sigma_prime_termination Is it true that iterates of $n\mapsto\sigma(n) - 1$ always reach a prime?
Erdos41.NtupleCondition NtupleCondition A n says that the sum of n elements of A determines the summands, counted with multiplicity and up to permutation
Erdos41.erdos_41 Let $A \subset \mathbb{N}$ be an infinite set such that the triple sums $a+b+c$ are all distinct for $a,b,c \in A$ (aside from the trivial coincidences)
Erdos41.erdos_41.variants.pairwise Erdős proved that if the pairwise sums
Erdos410.erdos_410 Let $σ_1(n) = σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$
Erdos412.erdos_412 Let $σ_1(n)=σ(n)$, the sum of divisors function, and $σ_k(n) = σ(σ_{k-1}(n))$
Erdos413.IsBarrier IsBarrier f n means n is a barrier for the real-valued function f, i.e. (m : ℝ) + f m ≤ (n : ℝ) for all m < n
Erdos413.erdos_413.parts.i Are there infinitely many barriers for ω?
Erdos413.expProd expProd n is ∏ kᵢ when n = ∏ pᵢ ^ kᵢ, i.e. the product of the prime exponents of n
Erdos413.erdos_413.variants.hasPosDensity_barrier_expProd Erdős proved that the barrier set for expProd is infinite and even has positive density
Erdos413.erdos_413.variants.bigOmega Erdős believed there should be infinitely many barriers for Ω, the total prime multiplicity
Erdos413.erdos_413.variants.bigOmega_largest_barrier_lt_100k Selfridge computed that the largest Ω-barrier below 10^5 is 99840
Erdos413.erdos_413.parts.ii Does there exist some $\epsilon > 0$ such that there are infinitely many $\epsilon$-barriers for $\omega$?
Erdos414.h The auxiliary function $h(n) = n + τ(n)$ (where $τ(n) counts the number of divisors of $n$)
Erdos414.erdos_414 Let $h_1(n) = h(n)$ and $h_k(n) = h(h_{k-1}(n))$
Erdos416.V Let V(x) count the number of n≤x such that ϕ(m)=n is solvable
Erdos416.erdos_416.parts.i Let V(x) count the number of n≤x such that ϕ(m)=n is solvable
Erdos416.erdos_416.parts.ii Let V(x) count the number of n≤x such that ϕ(m)=n is solvable
Erdos416.erdos_416.variants.Pillai Let V(x) count the number of n≤x such that ϕ(m)=n is solvable
Erdos416.erdos_416.variants.Erdos Let V(x) count the number of n≤x such that ϕ(m)=n is solvable
Erdos416.erdos_416.variants.Maier_Pomerance Let V(x) count the number of n≤x such that ϕ(m)=n is solvable
Erdos416.erdos_416.variants.Ford Let V(x) count the number of n≤x such that ϕ(m)=n is solvable
Erdos417.erdos_417.parts.i Let$$V'(x)=\#\{\phi(m) : 1\leq m\leq x\}$$and$$V(x)=\#\{\phi(m) \leq x : 1\leq m\}.$$ Does $\lim V(x)/V'(x)$ exist?
Erdos417.erdos_417.parts.ii Is it $>1$?
Erdos418.erdos_418 Are there infinitely many integers not of the form $n - \phi(n)$?
Erdos418.erdos_418.variants.conditional It follows from a slight strengthening of the Goldbach conjecture that every odd number can be written as $n - \phi(n)$
Erdos418.erdos_418.variants.odd_noncototient The Odd Noncototient Conjecture: every non-cototient is even
Erdos418.erdos_418.variants.seven_mem_cototient A sanity check for the definition: $7$ is a cototient, since $7 = 15 - \phi(15)$ and $\phi(15) = 8$
Erdos418.erdos_418.variants.sigma Documented declaration: sigma
Erdos418.erdos_418.variants.soln A solution to erdos_418 was shown by Browkin and Schinzel [BrSc95] by showing that any integer of the form $2^(k + 1)\cdot 509203$ is not of the form $n - \phi(n)$
Erdos418.erdos_418.variants.density It is open whether the set of non-cototients has positive (lower) density
Erdos419.factorialDivisorRatio The ratio $\sigma_0((n+1)!)/\sigma_0(n!)$, where $\sigma_0$ is the divisor-counting function
Erdos419.limitPointSet The set $\{1\} \cup \{1+1/k : k \geq 1\}$
Erdos419.erdos_419 Documented declaration: erdos_419
Erdos42.erdos_42 Erdős Problem 42: Let M ≥ 1 and N be sufficiently large in terms of M
Erdos42.example_maximal_sidon The set {1, 2, 4} is a maximal Sidon set in {1, ..., 4}
Erdos42.example_difference_set The difference set of {1, 2, 4} is {0, 1, 2, 3}
Erdos42.maximal_sidon_contains_zero For any maximal Sidon set, the difference set contains 0
Erdos422.IsHofstadterQ Documented declaration: IsHofstadterQ
Erdos422.erdos_422.test.f3 The third term of Hofstadter's $Q$-sequence is $f(3) = f(2) + f(2) = 2$
Erdos422.erdos_422.test.f4 The fourth term of Hofstadter's $Q$-sequence is $f(4) = f(2) + f(3) = 3$
Erdos422.erdos_422 Let $f(1) = f(2) = 1$ and for $n > 2$ $$ f(n) = f(n - f(n - 1)) + f(n - f(n - 2)). $$ Does $f(n)$ miss infinitely many integers?
Erdos422.erdos_422.variants.surjective Is $f$ surjective?
Erdos422.erdos_422.variants.growth_rate How does $f$ grow?
Erdos422.erdos_422.variants.eventually_const Does $f$ become stationary at some point?
Erdos423.IsConsecutiveBlockSum IsConsecutiveBlockSum a k m means that $m$ equals the sum of at least two consecutive terms of the sequence $a$, using indices from $\{1, \ldots, k - 1\}$
Erdos423.IsHofstadterSeq The Hofstadter sequence (OEIS A005243)
Erdos423.erdos_423.test.a3 The third term of the Hofstadter sequence is $a(3) = 3 = a(1) + a(2) = 1 + 2$
Erdos423.erdos_423.test.a4 The fourth term of the Hofstadter sequence is $a(4) = 5 = a(2) + a(3) = 2 + 3$
Erdos423.erdos_423 Documented declaration: erdos_423
Erdos423.erdos_423.variants.nondecreasing Bolan and Tang [Ta26] independently proved that $a_n-n$ is nondecreasing
Erdos423.erdos_423.variants.unbounded Bolan and Tang [Ta26] independently proved that $a_n-n\to\infty$
Erdos423.erdos_423.variants.infinite_complement Bolan and Tang [Ta26] independently proved that infinitely many positive integers do not occur in the Hofstadter sequence
Erdos423.IsHofstadterSeq.strictMono A Hofstadter sequence is strictly increasing from index 1 on
Erdos423.IsHofstadterSeq.le_apply A Hofstadter sequence satisfies n ≤ a n for n ≥ 1
Erdos423.IsHofstadterSeq.lt_iff_lt For indices ≥ 1, a Hofstadter sequence preserves and reflects <
Erdos423.IsHofstadterSeq.infinite_compl_of_unbounded Documented declaration: infinite_compl_of_unbounded
Erdos423.IsHofstadterSeq.unbounded_of_infinite_compl If infinitely many integers are missed, a n - n is unbounded: M + 1 missed integers below n together with a 1, …, a n are distinct elements of [1, a n]
Erdos423.erdos_423.test.unbounded_iff_infinite_complement The unboundedness of $a_n-n$ is equivalent to the sequence omitting infinitely many positive integers
Erdos423.erdos_423.variants.upper_bound Tang [Ta26] proved $a_n \ll n^{1/(c-1)+o(1)}$ whenever every finite convex set $A$ satisfies $|A-A|\geq |A|^{c-o(1)}$
Erdos423.erdos_423.variants.lower_bound Tang [Ta26] proved the lower bound $a_n=n+\Omega(\log\log n)$
Erdos424.nextGeneration Defines the set of new numbers generated from a set A by the operation $x y - 1$ for $x \neq y$
Erdos424.sequenceSet The sequence of sets $A_n$ where $A_0 = \{2, 3\}$ and $A_{n+1}$ is $A_n$ union all newly generated elements
Erdos424.generatedSet The set of integers which eventually appear in the sequence, which is the union of all $A_n$
Erdos424.erdos_424.variants.exact_density A literal interpretation of "positive density": the natural density of generatedSet exists (i.e. the lower and upper density agree) and is positive
Erdos426.isUniqueSubgraph_bot_bot Sanity check: the empty graph ⊥ is a unique subgraph of itself
Erdos426.erdos_426 We say $H$ is a unique subgraph of $G$ if there is exactly one way to find $H$ as a subgraph (not necessarily induced) of $G$
Erdos427.erdos427 The predicate that for every $n$ and $d$, there exists $k$ such that $$ d \mid p_{n + 1} + \cdots + p_{n + k}, $$ where $p_r$ denotes the $r$th prime?
Erdos427.ShiuTheorem Documented declaration: ShiuTheorem
Erdos427.erdos_427.variants.shiu Documented declaration: shiu
Erdos427.erdos_427.variants.of_shiu Cedric Pilatte has observed that a positive solution to Erdős Problem 427 follows from Shiu's theorem
Erdos428.primeDensityRatio The density ratio of set $A$ up to $n$ relative to the prime counting function $\pi(n)$
Erdos428.erdos_428 Is there a set
Erdos429.erdos_429 Documented declaration: erdos_429
Erdos43.f Let $f(N)$ be the maximum possible size of a Sidon set in $\{1,\ldots,N\}$
Erdos43.erdos_43.parts.i Documented declaration: i
Erdos43.erdos_43.parts.ii Documented declaration: ii
Erdos430.terms The terms of the sequence of Erdős Problem 430 for a given $n$: the integers $m$ with $1 < m < n$ all of whose prime factors are $> n - m$
Erdos430.terms_eight For example if $n=8$ we have $a_1=7$ and $a_2=5$ and then must stop
Erdos430.isGreatest_sub_one The first term of the sequence is $a_1 = n - 1$, the largest element of terms n
Erdos430.erdos_430 Fix some integer
Erdos431.erdos_431 Are there two infinite sets $A$ and $B$ such that $A+B$ agrees with the primes up to finitely many exceptions?
Erdos433.frobeniusNumber The Frobenius number of a finite set $A$: the greatest natural number outside the additive submonoid generated by $A$
Erdos433.g The largest Frobenius number of a $k$-element subset of $\{1,\ldots,n\}$ whose elements have greatest common divisor one
Erdos433.erdos_433 For every fixed $k \geq 2$, Erdős and Graham conjectured that $$g(k,n) \sim \frac{n^2}{k-1}.$$ The conjecture was proved by Dixmier [Di90]
Erdos434.Nat.IsRepresentableAs A natural $n$ is representable as a set $A$ if it can be written as the sum of finitely many elements of $A$ (with repetition allowed)
Erdos434.Nat.NcardUnrepresentable The number of naturals that cannot be written as the sum of finitely many elements of the set $A$, with repetition allowed
Erdos434.erdos_434.parts.i Let $k \le n$
Erdos434.erdos_434.parts.ii Documented declaration: ii
Erdos435.erdos_435 Let $n\in\mathbb{N}$ with $n\neq p^k$ for any prime $p$ and $k\geq 0$
Erdos437.squarePartialProducts The number of partial products $a_1, a_1a_2, \ldots, a_1\cdots a_k$ of a finite sequence a = [a₁, …, a_k] which are squares
Erdos437.L L x is the maximal number of square partial products of a sequence $1\leq a_1<\cdots<a_k\leq x$
Erdos437.erdos_437.variants.little_o Erdős and Graham write it is 'trivial' that there are $o(x)$ many such squares, although this is not quite trivial, using Siegel's theorem
Erdos437.erdos_437.variants.tao Tao [Ta24] shows that, if $u(x)=(\log x\log\log x)^{1/2}$, then $$x\exp(-(2^{1/2}+o(1))u(x)) \leq L(x) \leq x\exp(-(2^{-1/2}+o(1))u(x)).$$
Erdos438.SquareSumFree A finite set of natural numbers is square-sum-free if the sum of any two of its elements (possibly equal) is not a square
Erdos438.extremalSize The largest size of a square-sum-free subset of {1, …, N}
Erdos44.maxSidonSubsetCard_icc_bound The maximum size of a Sidon set in {1, ..., N} is less than or equal to 2 * √N
Erdos44.erdos_44 Erdős Problem 44: Let N ≥ 1 and A ⊆ {1,…,N} be a Sidon set
Erdos44.erdos_44.variants.empty_start The case where we start with an empty set (constructing large Sidon sets)
Erdos44.example_sidon_set The set {1, 2, 4, 8, 13} is a Sidon set in {1, ..., 13}
Erdos44.sidon_set_lower_bound For any N, there exists a Sidon set of size at least √N/2
Erdos44.greedy_sidon_construction The greedy construction gives a Sidon set A ⊆ {1, ..., N} of size at least $N^{1/3}$, stated here as $N \le |A|^3$
Erdos440.count For a strictly increasing sequence a of positive integers, count a x is the number of indices i with lcm(aᵢ, aᵢ₊₁) ≤ x
Erdos440.erdosSzemerediConstant The Erdős–Szemerédi constant $c=\sum_{n\geq 1}\frac{1}{n^{1/2}(n+1)}\approx 1.86$ (summed here over $n = m + 1$, $m \geq 0$)
Erdos441.g g N is the size of the largest A ⊆ {1, …, N} such that lcm(a, b) ≤ N for all a, b ∈ A
Erdos441.erdosConstruction Erdős' construction: all integers in $[1,(N/2)^{1/2}]$ together with all even integers in $[(N/2)^{1/2},(2N)^{1/2}]$
Erdos441.erdos_441.variants.chen_dai_infinitely_often Chen and Dai [ChDa07] proved that, infinitely often, Erdős' construction is not optimal: for infinitely many
Erdos441.erdos_441.variants.lower_bound Erdős' construction proves that $g(N) \geq \left(\tfrac{9}{8}N\right)^{1/2}+O(1)$
Erdos441.erdos_441.variants.erdos_upper_bound Erdős [Er51b] proved $g(N) \leq (4N)^{1/2}+O(1)$
Erdos441.erdos_441.variants.chen Chen [Ch98] established the asymptotic $g(N) \sim \left(\tfrac{9}{8}N\right)^{1/2}$
Erdos441.erdos_441.variants.chen_dai_upper_bound Chen and Dai [DaCh06] proved that $g(N)\leq \left(\tfrac{9}{8}N\right)^{1/2}+ O\left(\left(\frac{N}{\log N}\right)^{1/2}\log\log N\right)$
Erdos442.Real.maxLogOne The function $\operatorname{Log} x := \max\{log x, 1\}$
Erdos442.Set.bddProdUpper If A be a set of natural numbers and let x be real, then A.bddProdUpper x is the finite upper-triangular set of pairs of elements of A that are ≤ x
Erdos442.erdos_442.variants.tao Tao resolved erdos_442 in the negative in Theorem 1 of https://arxiv.org/pdf/2407.04226
Erdos443.A The set $\{k(m-k) : 1\leq k\leq m/2\}$, where m / 2 is ℕ floor division
Erdos443.erdos_443.parts.i Let $m,n\geq 1$
Erdos443.erdos_443.parts.ii Let $m,n\geq 1$
Erdos445.Erdos445Prop The property that there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$
Erdos445.erdos_445 Is it true that, for any $c>1/2$, if $p$ is a sufficiently large prime then, for any $n\geq 0$, there exist $a,b\in(n,n+p^c)$ such that $ab\equiv 1\pmod{p}$?
Erdos445.erdos_445.variants.heilbronn Heilbronn (unpublished) proved this for $c$ sufficiently close to $1$
Erdos445.erdos_445.variants.heath_brown Heath-Brown [He00] used Kloosterman sums to prove this for all $c>3/4$
Erdos445.erdos_445.test.small_example Small example: for $p=5$, $c=1$, $n=0$, the pair $(2,3) \in (0,5)$ satisfies $2 \cdot 3 = 6 \equiv 1 \pmod{5}$
Erdos447.maxUnionFree The largest size of a union-free collection $\mathcal{F}$ of subsets of $[n]$
Erdos447.erdos_447.parts.i How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be?
Erdos447.erdos_447.parts.ii How large can a union-free collection $\mathcal{F}$ of subsets of $[n]$ be?
Erdos448.tauPlus tauPlus n (written $\tau^+(n)$) counts the number of $k$ such that $n$ has a divisor in $[2^k, 2^{k+1})$
Erdos448.tauPlus_six Sanity check: $\tau^+(6) = 3$
Erdos448.tauPlus_twelve Sanity check: $\tau^+(12) = 4$
Erdos448.tauPlus_le_tau Always $\tau^+(n) \le \tau(n)$: the occupied dyadic blocks are the image of the divisor set under Nat.log 2, and an image has at most as many elements as its source
Erdos448.erdos_448.variants.erdos_tenenbaum Quantitative form of the (negative) answer to erdos_448
Erdos448.erdos_448.variants.hall_tenenbaum_upper_bound Documented declaration: hall_tenenbaum_upper_bound
Erdos448.erdos_448.variants.hall_tenenbaum_distribution Documented declaration: hall_tenenbaum_distribution
Erdos448.erdos_448.variants.ford Erdős and Graham asked whether there is a good inequality for $\sum_{n \le x} \tau^+(n)$
Erdos45.erdos_45 Let $k\geq 2$
Erdos450.HasMediumDivisor m has a divisor strictly between n and 2n
Erdos450.localCount The number of integers strictly between x and x + y with a divisor in (n, 2n)
Erdos450.UniformlySparse Every window (x, x+y) has at most ε y integers with a divisor in (n, 2n)
Erdos450.IsSufficientScale Y ε n is a sufficient window length: for every ε > 0, all large n, and every y ≥ Y ε n, the window is ε-sparse
Erdos450.windowThreshold The least window length y₀ such that every window (x, x+y) with y ≥ y₀ is ε-sparse, or ⊤ if no such y₀ exists
Erdos450.erdos_450 How large must $y=y(\epsilon,n)$ be such that the number of integers in $(x,x+y)$ with a divisor in $(n,2n)$ is at most $\epsilon y$?
Erdos450.erdos_450.linear_scale_suffices A translate-uniform linear scale suffices: there is a sufficient window length Y with Y ε n ≤ C(ε) · n
Erdos452.largeOmegaIntervalLength The greatest length of an interval in $[x,2x]$ on which $\omega(n) > \log\log n$ everywhere
Erdos452.erdos_452 Determine the largest length of an interval in $[x,2x]$ on which $\omega(n) > \log\log n$ everywhere
Erdos453.EventuallyHasPrimeWitness The eventual prime inequality asked in Erdős Problem 453, using Mathlib's zero-based primes
Erdos453.erdos_453 Is it true that, for all sufficiently large $n$, there exists some $i<n$ such that $$ p_n^2 < p_{n+i}p_{n-i}, $$ where $p_k$ is the $k$th prime?
Erdos454.f Define f n to be the minimum of (n + i).nth Prime + (n - i).nth Prime over 0 < i < n
Erdos454.erdos_454 Is it true that limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop = ⊤?
Erdos454.erdos_454.variants.two_le_limsup limsup (fun n => (f n - 2 * n.nth Prime : ℕ∞)) atTop ≥ 2, and this is proved in [Po79]
Erdos455.erdos_455 Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n
Erdos455.erdos_455.variants.liminf Let q : ℕ → ℕ be a strictly increasing sequence of primes such that q (n + 2) - q (n + 1) ≥ q (n + 1) - q n
Erdos456.p Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$
Erdos456.m Let $m_n$ be the smallest integer such that $n\mid \phi(m_n)$
Erdos456.erdos_456.parts.i Is it true that $m_n<p_n$ for almost all $n$?
Erdos456.erdos_456.parts.ii Does $p_n/m_n \to \infty$ for almost all $n$?
Erdos456.erdos_456.parts.iii Are there infinitely many primes $p$ such that $p-1$ is the only $n$ for which $m_n=p$?
Erdos456.erdos_456.variants.linniks_theorem Linnik's theorem implies that $p_n\leq n^{O(1)}$
Erdos456.erdos_456.variants.mn_leq_pn It is trivial that $m_n \leq p_n$ always
Erdos456.erdos_456.variants.infinitely_many_n Erdős [Er79e] writes it is 'easy to show' that for infinitely many $n$ we have $m_n < p_n$
Erdos456.erdos_456.variants.m_div_n Erdős [Er79e] writes it is 'easy to show' that $m_n/n \to \infty$ for almost all $n$
Erdos457.erdos_457 Is there some
Erdos457.q Let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$
Erdos457.erdos_457.variants.qnk More generally, let $q(n, k)$ denote the least prime which does not divide $\prod_{1 \le i \le k}(n + i)$
Erdos457.erdos_457.variants.one_sub Taking
Erdos458.lcm_upto The least common multiple of the integers in the set $\{1, \dots, n\}$
Erdos458.erdos_458 Let $\operatorname{lcm}(1, \dots, n)$ denote the least common multiple of $\{1, \dots, n\}$
Erdos459.f The function from the problem, in its equivalent form: f u is the smallest v > u all of whose prime factors divide u
Erdos459.erdos_459 Let $f(u)$ be the largest $v$ such that no $m\in (u,v)$ is composed entirely of primes dividing $uv$
Erdos459.erdos_459.variants.upper_tight The upper bound $f u ≤ u ^ 2$ is attained exactly when u is prime: $f p = p ^ 2$
Erdos46.erdos_46 Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?
Erdos46.erdos_46.variants.infinitely_many_disjoint Croot [Cr03] proved more: there are infinitely many disjoint such monochromatic solutions
Erdos46.erdos_46.variants.positive_rat In [ErGr80] they also ask for a monochromatic representation of any $\frac{a}{b}>0$
Erdos462.erdos_462 Let $p(n)$ denote the least prime factor of $n$
Erdos463.erdos_463 Is there a function
Erdos464.erdos_464 Let $A=\{n_10$ with $n_{k+1}\geq (1+\epsilon)n_k$ for all $k$)
Erdos466.N N X δ is the maximum number of points in a closed disc of radius X in the plane such that the distance between any two of them is at least δ away from the nearest integer
Erdos466.erdos_466.variants.graham Graham proved that $N(X,1/10)> \frac{\log X}{10}$
Erdos466.erdos_466.variants.sarkozy Sárközy [Sa76] proved that for all sufficiently small $\delta>0$, $N(X,\delta)>X^{1/2-\delta^{1/7}}$ for all sufficiently large $X$
Erdos469.Nat.IsSumDivisors The proposition that n is a sum of distinct proper divisors
Erdos469.erdos_469 Let $A$ be the set of all $n$ such that $n = d_1 + ⋯ + d_k$ with $d_i$ distinct proper divisors of $n$, but this is not true for any $m ∣ n$ with $m < n$
Erdos470.PrimitiveWeird Primitive weird numbers are weird numbers such that no proper divisor of $n$ are weird
Erdos470.AbundancyIndex The abundancy index is the sum of the divisors of $n$ divided by $n$
Erdos470.erdos_470.parts.i Are there any odd weird numbers?
Erdos470.erdos_470.parts.ii Are there infinitely many primitive weird numbers?
Erdos470.erdos_470.variants.weird_pos_density Benkoski and Erdős BeEr74 proved that the set of weird numbers has positive density
Erdos470.erdos_470.variants.smallest_weird_eq_70 The smallest weird number is 70
Erdos470.erdos_470.variants.prime_gap_imp_inf_prim_weird Documented declaration: prime_gap_imp_inf_prim_weird
Erdos470.erdos_470.variants.odd_weird_10_pow_21 Fang Fa22 has shown there are no odd weird numbers below $10^{21}$
Erdos470.erdos_470.variants.odd_weird_prime_div Liddy and Riedl LiRi18 have shown that an odd weird number must have at least 6 prime divisors
Erdos473.erdos_473.variants.finite Segal also asked whether for every $n \ge 2$ there is a permutation $a_1, \ldots, a_n$ of $\{1, \ldots, n\}$ such that $a_k + a_{k+1}$ is prime for all $1 \le k < n$
Erdos476.erdos_476 Let $A\subseteq \mathbb{F}_p$
Erdos477.erdos_477.variants.S_sq There is no such $A$ for the polynomial $f(x) = X^2$
Erdos477.erdos_477.variants.degree_two_dvd_condition_b_ne_zero There is no such $A$ for any polynomial $f(x) = aX^2 + bX + c$, if $a | b$ with $a \ne 0$ and $b \ne 0$
Erdos477.erdos_477.variants.X_pow_three Probably there is no such $A$ for the polynomial $X^3$
Erdos478.erdos_478 Let $p$ be a prime and $$A_p = \{ k! \pmod{p} : 1\leq k<p\}.$$ Is it true that $$\lvert A_p\rvert \sim (1-\tfrac{1}{e})p?$$
Erdos479.erdos_479 Is it true that, for every integer $k\neq 1$, there are infinitely many $n$ such that $2^n\equiv k\pmod{n}$?
Erdos480.erdos_480 Let $x_1,x_2,\ldots\in [0,1]$ be an infinite sequence
Erdos480.erdos_480.variants.chung_graham Documented declaration: chung_graham
Erdos480.erdos_480.variants.chung_graham_best_possible They also prove that this constant is best possible
Erdos481.T Documented declaration: T
Erdos481.erdos_481 Let $a_1,\ldots,a_r,b_1,\ldots,b_r\in \mathbb{N}$ such that $\sum_{i}\frac{1}{a_i}>1$
Erdos482.a The sequence of the problem: $a_1 = 1$ and $a_{n+1} = \lfloor \sqrt{2}(a_n + 1/2) \rfloor$ for $n \geq 1$
Erdos482.generalRecurrence Stoll's general Graham–Pollak-type recurrence in base $g$, with real parameters $a$, $b$, and $\varepsilon$
Erdos482.alpha The odd-step coefficient of the Rabinowitz–Gilbert/Stoll binary recurrence
Erdos482.beta The even-step coefficient of the Rabinowitz–Gilbert/Stoll binary recurrence
Erdos482.stollBinary Zero-based form of the Rabinowitz–Gilbert/Stoll binary recurrence: stollBinary t k is the paper's $u_{k+1}$
Erdos482.binaryDigit The binary digits of a normalized real $t \in [1, 2)$, indexed from one and including the leading digit
Erdos484.erdos_484 Prove that there exists an absolute constant
Erdos485.f The minimum number of terms of the square of a rational polynomial with exactly k nonzero terms, where the number of terms of P is P.support.card
Erdos487.erdos_487 Let $A\subseteq \mathbb{N}$ have positive density
Erdos488.erdos_488 Documented declaration: erdos_488
Erdos489.sievedSet The set of positive integers not divisible by any element of A
Erdos489.GapSumSq The squared-gap sum ∑_{b_i < x} (b_{i+1} - b_i)², where b_i enumerates the positive integers not divisible by any element of A
Erdos489.erdos_489 Let $A\subseteq \mathbb{N}$ be a set such that $\lvert A\cap [1,x]\rvert=o(x^{1/2})$
Erdos489.erdos_489.variants.squarefree When $A = \{p^2 : p \textrm{ prime}\}$, $B$ is the set of squarefree numbers, and the existence of this limit was proved by Erdős
Erdos49.f The largest size of a set $A\subseteq\{1,\ldots,N\}$ on which Euler's totient function is strictly increasing
Erdos49.g The largest size of a set $A\subseteq\{1,\ldots,N\}$ on which Euler's totient function is non-decreasing
Erdos49.erdos_49.variants.primes_maximal Are the primes the largest possible example, i.e. is $f(N) = \pi(N)$ for all $N \geq 2$?
Erdos490.erdos_490.variants.limit Erdős [Er72] asks whether
Erdos493.erdos_493 Does there exist a $k$ such that every sufficiently large integer can be written in the form $$\prod_{i=1}^k a_i - \sum_{i=1}^k a_i$$ for some integers $a_i\geq 2$?
Erdos494.sumMultiset For a finite set $A \subset \mathbb{C}$ and $k \ge 1$, define $A_k$ as the multiset consisting of all sums of $k$ distinct elements of $A$
Erdos494.erdos_494.variants.k_eq_2_card_not_pow_two Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 2$ and $|A| \ne 2^l$ for $l \ge 0$
Erdos494.erdos_494.variants.k_eq_3_card_gt_6 Selfridge and Straus [SeSt58] also showed that the conjecture is true when 1) $k = 3$ and $|A| > 6$, except possibly for $|A| = 27$ and $|A| = 486$, or 2) $k = 4$ and $|A| > 12$
Erdos494.erdos_494.variants.k_eq_4_card_gt_12 Selfridge and Straus [SeSt58] showed that the conjecture is true when $k = 4$ and $|A| > 12$
Erdos494.erdos_494.variants.card_divisible_by_prime_gt_k Selfridge and Straus [SeSt58] proved that $A$ is determined by $A_k$ if $|A|$ is divisible by a prime greater than $k$
Erdos494.erdos_494.variants.gordon_fraenkel_straus Gordon, Fraenkel, and Straus [GRS62] proved that the claim is true for all $k > 2$ when $|A|$ is sufficiently large
Erdos494.prodMultiset A version in [Er61] by Erdős is product instead of sum, which is false
Erdos495.erdos_495 Let $\alpha,\beta \in \mathbb{R}$
Erdos497.erdos_497 How many antichains in $[n]$ are there?
Erdos498.erdos_498 Let $z_1,\ldots,z_n\in\mathbb{C}$ with $1\leq \lvert z_i\rvert$ for $1\leq i\leq n$
Erdos499.erdos_499 Let $M$ be a real $n \times n$ doubly stochastic matrix
Erdos499.vanDerWaerden The conjecture of van der Waerden, which states that the permanent of a doubly stochastic matrix is at least $n^{-n} n!$
Erdos499.erdos_499.variants.one_le A weaker version of Erdős' problem 499, which asks whether for every doubly stochastic matrix, there exists a permutation
Erdos5.normalizedGap The normalised prime gap $\frac{p_{n+1}-p_n}{\log n}$, where $p_n$ denotes the $n$-th prime
Erdos5.limitPointSet The set $S$ of limit points of $\frac{p_{n+1}-p_n}{\log n}$
Erdos5.erdos_5 Let $C\geq 0$
Erdos5.erdos_5.variants.limit_point_set Let $S$ be the set of limit points of $(p_{n+1}-p_n)/\log n$
Erdos5.erdos_5.variants.westzynthius $\infty\in S$ by Westzynthius' result [We31] on large prime gaps
Erdos5.erdos_5.variants.goldston_pintz_yildirim $0\in S$ by the work of Goldston, Pintz, and Yildirim [GPY09] on small prime gaps
Erdos5.erdos_5.variants.erdos_ricci Erdős [Er55] and Ricci [Ri56] independently showed that $S$ has positive Lebesgue measure
Erdos5.erdos_5.variants.hildebrand_maier Hildebrand and Maier [HiMa88] showed that $S$ contains arbitrarily large (finite) numbers
Erdos5.erdos_5.variants.hildebrand_maier_measure [HiMa88] in fact prove the stronger statement that there is a constant $c>0$ with $\lambda([0,T]\cap S)\geq cT$ for all sufficiently large $T$
Erdos5.erdos_5.variants.pintz Pintz [Pi16] showed that there exists some small constant $c>0$ such that $[0,c]\subset S$
Erdos5.erdos_5.variants.banks_freiberg_maynard Banks, Freiberg, and Maynard [BFM16] showed that at least $12.5\%$ of $[0,\infty)$ belongs to $S$
Erdos5.erdos_5.variants.banks_freiberg_maynard_measure The $12.5\%$ claim itself, as deduced in [BFM16, Corollary 1.2]: $\lambda([0,T]\cap S)\geq (1-o(1))T/8$ as $T\to\infty$
Erdos5.erdos_5.variants.merikoski Merikoski [Me20] showed that at least $1/3$ of $[0,\infty)$ belongs to $S$
Erdos5.erdos_5.variants.merikoski_measure The $1/3$ claim itself [Me20, Corollary 2]: $\lambda([0,T]\cap S)\geq T/3$ for all $T>0$
Erdos5.erdos_5.variants.merikoski_bounded_gaps Merikoski [Me20] showed that $S$ has bounded gaps
Erdos5.erdos_5.variants.dense In [Er65b], [Er85c], and [Er97c] Erdős asks whether
Erdos5.mem_limitPointSet_iff Membership in limitPointSet is exactly the existence of an infinite sequence of indices along which the normalised prime gaps converge, as in the statement of erdos_5
Erdos5.normalizedGap_nonneg The normalised prime gaps are nonnegative
Erdos5.limitPointSet_subset_Ici Every limit point of the normalised prime gaps is nonnegative
Erdos5.erdos_5_iff_limit_point_set Documented declaration: erdos_5_iff_limit_point_set
Erdos5.isClosed_limitPointSet Documented declaration: isClosed_limitPointSet
Erdos5.dense_iff_limit_point_set Weisenberg's remark, as reported on erdosproblems.com/5: since
Erdos50.IsDistributionOfPhiRatio Documented declaration: IsDistributionOfPhiRatio
Erdos50.IsPurelySingularOn Documented declaration: IsPurelySingularOn
Erdos50.erdos_50_schoenberg Schoenberg [Sch38] proved that the asymptotic distribution function of $\varphi(n)/n$ exists
Erdos50.erdos_50_singular Documented declaration: erdos_50_singular
Erdos50.erdos_50 Let $f$ be the asymptotic distribution function of $\varphi(n)/n$, so that for each $c \in [0,1]$, $f(c)$ is the natural density of $\{n : \varphi(n) < cn\}$
Erdos500.erdos_500 What is $\mathrm{ex}_3(n,K_4^3)$?
Erdos501.erdos_501 For every $x \in \mathbb{R}$ let $A_x \subset \mathbb{R}$ be a bounded set with outer measure $< 1$
Erdos501.erdos_501.variants.erdosHajnal_finite Erdős–Hajnal (1960): arbitrarily large finite independent sets exist
Erdos501.erdos_501.variants.hechler_CH Hechler (1972) [He72]: the answer to the main question is NO, assuming the continuum hypothesis
Erdos501.erdos_501.variants.gladysz_size2 Gladysz (1962) [Gl62]: independent set of size 2 in the closed case
Erdos501.erdos_501.variants.singleton_independent Trivial lower bound: a single-element set is always independent
Erdos501.erdos_501.variants.pair_independent_iff Two-element sets: independent iff mutual non-membership
Erdos501.erdos_501.tests.empty_family_is_valid The constant family A x = ∅ satisfies all hypotheses of the main problem: each A x is bounded (the empty set is bounded) and has Lebesgue outer measure 0 < 1
Erdos501.erdos_501.tests.singleton_zero_independent A singleton {0} is an independent set for any family A : ℝ → Set ℝ, as witnessed by erdos_501.variants.singleton_independent
Erdos501.erdos_501.tests.pair_independent_empty Two reals form an independent set for the empty family A _ = ∅: neither 0 nor 1 belongs to ∅, so both conditions of pair_independent_iff hold
Erdos501.erdos_501.tests.singleton_outer_measure_lt_one The hypothesis volume.toOuterMeasure (A x) < 1 is strictly satisfied when A x = {x} (a singleton), since Lebesgue measure of a singleton is 0
Erdos501.erdos_501.tests.unit_interval_measure The boundary case: the measure condition < 1 is sharp
Erdos502.erdos_502 What is the size of the largest $A\subseteq \mathbb{R}^n$ such that there are only two distinct distances between elements of $A$?
Erdos503.erdos_503 What is the size of the largest $A \subseteq \mathbb{R}^n$ such that every three points from $A$ determine an isosceles triangle?
Erdos503.erdos_503.variants.R2 When $n = 2$, the answer is 6 (due to Kelly [ErKe47] - an alternative proof is given by Kovács [Ko24c])
Erdos503.erdos_503.variants.R3 When $n = 3$, the answer is 8 (due to Croft [Cr62])
Erdos503.erdos_503.variants.upper_bound The best upper bound known in general is due to Blokhius [Bl84] who showed that $$ |A| \leq \binom{n + 2}{2} $$ [Bl84] Blokhuis, A., Few-distance sets
Erdos503.erdos_503.variants.lower_bound Documented declaration: lower_bound
Erdos505.erdos_505.small_dim Borsuk's conjecture, small dimensions: Borsuk.BorsukConjecture n holds for $n \leq 3$
Erdos506.numCircles The number of circles determined by a set of points P in the plane: the number of distinct circles (spheres) passing through at least three of the points of P
Erdos506.erdos_506 What is the minimum number of circles determined by any $n$ points in $\mathbb{R}^2$, not all on a circle?
Erdos506.erdos_506.variants.large_n Documented declaration: large_n
Erdos506.erdos_506.variants.small_n The problem appears to remain open for small
Erdos506.erdos_506.variants.segre_n_eq_eight Segre's observation: the lower bound $\binom{n-1}{2}$ (without the correction of [PuSm]) is already false for $n = 8$, as witnessed by the projection of a cube onto a plane
Erdos507.minTriangleArea The minimum (unsigned) area of a triangle determined by three distinct points in a set S
Erdos507.α $\alpha(n)$ is the supremum of minTriangleArea S over all sets S of $n$ points in the unit disk
Erdos507.lowerBest Current best lower bound [KPS82]
Erdos507.upperBarrier The "Barrier" function: n^(-7/6) used for the best upper bound [CPZ24]
Erdos507.erdos_507.equivalent Let $\alpha(n)$ be such that every set of $n$ points in the unit disk contains three points which determine a triangle of area at most $\alpha(n)$
Erdos507.erdos_507.lower Estimate a lower bound for$\alpha(n)$
Erdos507.erdos_507.upper Estimate an upper bound for$\alpha(n)$
Erdos507.erdos_507.variants.upper_trivial It is trivial that $\alpha(n) \ll 1/n$
Erdos507.erdos_507.variants.lower_erdos Erdős observed that $\alpha(n) \gg 1/n^2$
Erdos507.erdos_507.variants.lower_kps82 Current best lower bound [KPS82]
Erdos507.erdos_507.variants.upper_cpz24 Current best upper bound [CPZ24]: $\alpha(n) \ll n^{-7/6 + o(1)}$
Erdos508.HadwigerNelsonProblem The Hadwiger–Nelson problem asks: How many colors are required to color the plane such that no two points at distance 1 from each other have the same color?
Erdos508.HadwigerNelsonAtLeastFive Aubrey de Grey improved the lower bound for the chromatic number of the plane to 5 in 2018 using a graph that has >1000 nodes
Erdos508.HadwigerNelsonAtLeast4 The "chromatic number of the plane" is at least 4
Erdos508.HadwigerNelsonAtMostSeven This upper bound for the chromatic number of the plane was observed by John R
Erdos508.HadwigerNelsonAtLeastThree The chromatic number of the plane is at least 3
Erdos509.BoundedDiscCover An $r$-bounded disc cover of a subset of a metric space $M$ is an indexed family of closed discs whose radii sum to at most $r$
Erdos509.erdos_509 Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial
Erdos509.erdos_509.variants.Cartan_bound Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial
Erdos509.erdos_509.variants.Pommerenke_bound Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial
Erdos509.erdos_509.variants.Pommerenke_connected Let $f(z) ∈ ℂ[z]$ be a monic non-constant polynomial
Erdos51.erdos_51 Is there an infinite set
Erdos510.erdos_510 Documented declaration: erdos_510
Erdos510.erdos_510.variants.ruzsa Ruzsa [Ru04] proved an upper bound of $-\exp(O(\sqrt{\log N})$
Erdos510.erdos_510.variants.bedert Bedert [Be25c] proved an upper bound of $-c N^{1/7}$
Erdos512.erdos_512 Documented declaration: erdos_512
Erdos513.erdos_513 Let f be a transcendental entire function
Erdos513.erdos_513.variants.upper_bound For all transcendental entire function f, liminf (fun r : ℝ => ratio r f) atTop ≤ 2 / π - c for some c > 0
Erdos513.erdos_513.variants.lower_bound For all transcendental entire function f, liminf (fun r : ℝ => ratio r f) atTop > 1 / 2
Erdos516.OfFiniteOrder An entire function f is said to be of finite order if there exist numbers c, a ≥ 0 such that for all z, ‖f z‖ ≤ c * rexp (‖z‖ ^ a)
Erdos516.erdos_516.variants.limsup_ratio_eq_one Let f = ∑ aₖzⁿₖ be an entire function with strictly increasing exponents such that nₖ > k (log k) ^ (2 + c)
Erdos517.erdos_517 If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞, is it true that f assumes every value infinitely often?
Erdos517.erdos_517.variants.fejer If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that ∑ 1 / nₖ < ∞, then f assumes every value infinitely often
Erdos519.powerSum The $k$th power sum of a finite sequence of complex numbers
Erdos519.erdos_519 Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$
Erdos52.erdos_52 Let $A$ be a finite set of integers
Erdos520.IsRademacherMultiplicative A random function
Erdos521.sign true encodes the sign +1, false the sign -1
Erdos521.fairCoin One fair coin: the uniform probability measure on Bool
Erdos521.fairCoin_apply fairCoin gives each of the two signs mass 1/2
Erdos521.fairCoin_isProbabilityMeasure fairCoin is a probability measure
Erdos521.rademacherMeasure The law of an infinite sequence of independent fair coins (ε_k)_{k ≥ 0}, each uniform on {-1, +1}
Erdos521.littlewoodPolynomial The degree-n Littlewood polynomial f_n(z) = ∑_{0 ≤ k ≤ n} ε_k z^k
Erdos521.realRootCount R_n: the number of distinct real roots of f_n
Erdos521.Claim The assertion asked about in Problem 521: almost surely R_n / log n → 2/π
Erdos521.erdos_521 Let $(\epsilon_k)_{k\geq 0}$ be independently uniformly chosen at random from $\{-1,1\}$
Erdos522.KacCoefficients Documented declaration: KacCoefficients
Erdos522.KacCoefficients.polynomial The random polynomial associated to a sequence c : KacCoefficients S Ω μ of Kac coefficients given by ∑ i ∈ Finset.range (n + 1), c i z^i
Erdos522.KacCoefficients.roots The random multiset of roots associated to a Kac polynomial
Erdos522.KacCoefficients.numRootsInUnitDisk Counts the number of roots of a Kac polynomial in the unit disk with multiplicity
Erdos522.erdos_522.variants.number_real_roots Erdős and Offord showed that the number of real roots of a random degree n polynomial with ±1 coefficients is (2/π+o(1))log n
Erdos522.erdos_522.variants.yakir_solution Yakir proved that almost all Kac polynomials have n/2+O(n^(9/10)) many roots in {z∈C:|z|≤1}
Erdos525.eval The degree-n polynomial $f(z)=\sum_{j\leq n}\epsilon_j z^j$ with coefficients ε : Fin (n + 1) → ℤˣ, i.e. $\epsilon_j\in\{-1,1\}$, evaluated at z
Erdos525.exceptional The degree-n polynomials with $\pm 1$ coefficients such that $|f(z)|\geq 1$ for all $|z|=1$
Erdos525.m $m(f)=\min_{|z|=1}|f(z)|$
Erdos525.prob The probability that a uniformly random degree-n polynomial with $\pm 1$ coefficients satisfies P
Erdos525.erdos_525.variants.kashin Littlewood [Li66] conjectured that $m(f)=o(1)$ holds almost surely; this was proved by Kashin [Ka87]
Erdos525.erdos_525.variants.konyagin Konyagin [Ko94] proved that $m(f)\leq n^{-1/2+o(1)}$ almost surely
Erdos525.erdos_525.variants.konyagin_schlag Konyagin and Schlag [KoSc99] proved that for any $\epsilon>0$ $$\limsup_{n\to \infty} \mathbb{P}(m(f) \leq \epsilon n^{-1/2})\ll \epsilon.$$
Erdos53.sumsAndProducts The integers which are either the sum or the product of (one or more) distinct elements of A
Erdos53.erdos_53.variants.upper Erdős and Szemerédi [ErSz83] proved that there exist arbitrarily large sets
Erdos532.erdos_532 Documented declaration: erdos_532
Erdos533.erdos_533.variants.ehsss_upper The upper bound
Erdos533.erdos_533.variants.lrss_lower The matching lower bound
Erdos533.erdos_533.variants.delta_four_eq_zero The contrasting positive result $\delta_3(4) = 0$ of Erdős, Hajnal, Simonovits, Sós, and Szemerédi [EHSSS94]: the $K_4$ analogue of erdos_533 is true
Erdos533.erdos_533.variants.delta_seven_ge_quarter Documented declaration: delta_seven_ge_quarter
Erdos533.erdos_533.variants.test_bot Sanity check for the triangle-free-set condition: in the empty graph every vertex set spans no triangle, since the empty graph has no $3$-clique
Erdos534.IsAdmissible A set A ⊆ {1, …, N} is admissible if it contains N and any two distinct elements of A have a common factor
Erdos534.maxCard The largest size of an admissible subset of {1, …, N}
Erdos534.primePrefix The prime factors q₁ < ⋯ < qⱼ of N which are at most q
Erdos534.ahlswedeKhachatrian Documented declaration: ahlswedeKhachatrian
Erdos535.NoConstantPairwiseGcdCoprimeSubsets No r-subset has constant pairwise GCD with coprime quotients
Erdos535.AllBigOmega All elements of A are positive and have exactly k prime factors, counted with multiplicity
Erdos535.f f r N is the maximum size of a subset A ⊆ {1,…,N} such that no r-element subset of A has constant pairwise GCD
Erdos535.erdos_535 Documented declaration: erdos_535
Erdos535.erdos_535.variants.first_open_case The first open case of Erdős Problem 535 is $r = 3$: there should exist $c > 0$ such that $f_3(N) \leq N^{c/\log\log N}$ for all sufficiently large $N$
Erdos535.erdos_535.variants.erdos_upper_bound Erdős [Er64] proved that $f_r(N) \leq N^{3/4+o(1)}$ for all $r \geq 3$
Erdos535.erdos_535.variants.lower_bound Erdős [Er64] proved that $f_3(N) > N^{c/\log\log N}$ for some constant $c > 0$
Erdos535.erdos_535.variants.abbott_hanson Abbott and Hanson [AbHa70] improved Erdős's upper bound to $f_r(N) \leq N^{1/2+o(1)}$ for all $r \geq 3$
Erdos535.erdos_535.variants.sunflower_strong Erdős [Er73] records that Abbott pointed out the ordinary sunflower conjecture does not seem to suffice here
Erdos535.erdos_535.variants.sunflower_erdos_rado For the stronger $Ω(n)=k$ variant above, the Erdős–Rado method gives the weaker bound $c_r^k \cdot k!$; see Erdős [Er73]
Erdos536.erdos_536 Let $\epsilon>0$ and $N$ be sufficiently large
Erdos537.erdos_537 Let $\epsilon>0$ and $N$ be sufficiently large
Erdos538.representations The representations m = p * a with p prime and a ∈ A
Erdos538.Admissible A ⊆ {1, …, N} and every m has at most r representations m = p a
Erdos538.reciprocalMass The reciprocal sum ∑_{n ∈ A} 1/n of the problem
Erdos538.maxMass The largest reciprocal sum ∑_{n ∈ A} 1/n over admissible A ⊆ {1, …, N}
Erdos538.erdos_538 Let $r\geq 2$ and suppose that $A\subseteq\{1,\ldots,N\}$ is such that, for any $m$, there are at most $r$ solutions to $m=pa$ where $p$ is prime and $a\in A$
Erdos538.erdos_538.matching_order The reciprocal sum has matching order Θ_r(log N / loglog N): an explicit upper bound for every admissible A, together with a witnessing construction achieving the same order
Erdos539.IsCofactorLowerBound We say that $m$ is a cofactor lower bound for a given $n$ if, for every set $A$ of $n$ non-negative integers, there are at least $m$ cofactors $a / (a, b)$, where $a, b\in A$
Erdos539.cofactorThreshold The cofactor threshold $h(n)$, for every positive $n$, is the largest cofactor lower bound for $n$
Erdos539.erdos_539 Let $h(n)$ be maximal such that, for any set $A\subseteq \mathbb{N}$ of size $n$, the set$$\left\{ \frac{a}{(a,b)}: a,b\in A\right\}$$has size at least $h(n)$
Erdos539.erdos_539.variants.sq_isBigO Erdős and Szemerédi proved that$$n^{1/2} \ll h(n)$$
Erdos539.erdos_539.variants.isBigO_sq_cube_root Granville and Roesler [GR99] showed that $$h(n)\ll n^{2/3}$$
Erdos540.HasZeroSubsetSum A finite set has a non-empty subset whose sum is zero
Erdos540.erdos_540 Is it true that if $A\subseteq \mathbb{Z}/N\mathbb{Z}$ has size $\gg N^{1/2}$ then there exists some non-empty $S\subseteq A$ such that $\sum_{n\in S}n\equiv 0\pmod{N}$?
Erdos541.erdos_541 Documented declaration: erdos_541
Erdos541.erdos_541.variants.general_moduli Gao, Hamidoune, and Wang [GHW10] solved this for all moduli p (not necessarily prime)
Erdos541.erdos_541.variants.large_primes This was proved by Erdős and Szemerédi [ErSz76] for p sufficiently large
Erdos542.IsLcmFree A set A ⊆ {1, …, n} such that lcm(a, b) > n for all distinct a, b ∈ A
Erdos542.uncovered The integers 1 ≤ m ≤ n which are not divisible by any element of A
Erdos542.erdos_542.variants.log_power Schinzel and Szekeres [ScSz59] proved that there are examples $A\subseteq\{2,\ldots,n\}$ with at most $n/(\log n)^c$ many such $m$, for some real constant $c>0$
Erdos542.erdos_542.variants.chen Chen [Ch96] has proved that if $n>172509$ then $$\sum_{a\in A}\frac{1}{a}< \frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}.$$
Erdos542.erdos_542.variants.one_add_little_o In [Er73] Erdős further speculates that in fact $$\sum_{a\in A}\frac{1}{a}\leq 1+o(1),$$ where the $o(1)$ term $\to 0$ as $n\to \infty$
Erdos542.erdos_542.variants.only_two In [Er98] Erdős mentions that he, Schinzel, and Szekeres conjectured that $2,3,5$ and $3,4,5,7,11$ are the only two sequences for which the sum is $>1$
Erdos544.erdos_544.parts.i Show that $$R(3,k+1)-R(3,k)\to\infty$$ as $k\to \infty$
Erdos544.erdos_544.parts.ii Similarly, prove or disprove that $$R(3,k+1)-R(3,k)=o(k).$$
Erdos545.knPlusTEdges The graph formed by connecting a new vertex (none) to $t$ of the vertices of $K_n$ (Fin n)
Erdos545.erdos_545 Let $m$ be sufficiently large and let $G$ be a graph with $m$ edges and no isolated vertices
Erdos546.erdos_546 Let $G$ be a graph with no isolated vertices and $m$ edges
Erdos547.erdos_547 If $T$ is a tree on $n$ vertices then $$R(T) \leq 2n-2.$$ This problem is #14 in Ramsey Theory in the graphs problem collection
Erdos547.erdos_547.variants.sufficiently_large For sufficiently large $n$, every tree $T$ on $n$ vertices satisfies $R(T)\leq 2n-2$
Erdos549.erdos_549 Documented declaration: erdos_549
Erdos551.erdos_551 Prove that $$R(C_k,K_n)=(k-1)(n-1)+1$$ for $k\geq n\geq 3$ (except when $n=k=3$)
Erdos551.erdos_551.variants.sufficiently_large For sufficiently large $n$ and every $k\geq n$, $R(C_k,K_n)=(k-1)(n-1)+1$
Erdos552.erdos_552.parts.i Determine the Ramsey number $$R(C_4, S_n),$$ where $S_n=K_{1,n}$ is the star on $n+1$ vertices
Erdos552.erdos_552.parts.ii In particular, is it true that, for any $c > 0$, there are infinitely many $n$ such that $$R(C_4, S_n) \leq n + \sqrt{n} - c?$$
Erdos559.erdos_559.variants.rodl_szemeredi Rödl and Szemerédi [RoSz00] constructed, for infinitely many
Erdos559.erdos_559.variants.tikhomirov Tikhomirov [Ti22b] improved this to $\hat{R}(G)\gg n\exp(c\sqrt{\log n})$
Erdos559.erdos_559.variants.trees Friedman and Pippenger [FrPi87] proved the linear bound when $G$ is a tree
Erdos559.erdos_559.variants.draganic_petrova The best known upper bound for graphs of maximum degree $3$ is $\hat{R}(G)\leq n^{3/2+o(1)}$, due to Draganić and Petrova [DrPe22]
Erdos56.WeaklyDivisible Say a set of natural numbers is k-weakly divisible if any k+1 elements of A are not relatively prime
Erdos56.weaklyDivisible_singleton A singleton is k-weakly divisible if k ≠ 0
Erdos56.not_weaklyDivisible_zero No non-empty set is 1-weakly divisible
Erdos56.MaxWeaklyDivisible MaxWeaklyDivisible N k is the size of the largest k-weakly divisible subset of {1,..., N}
Erdos56.FirstPrimesMultiples FirstPrimesMultiples N k is the set of numbers in {1,..., N} that are a multiple of one of the first k primes
Erdos56.weaklyDivisible_firstPrimesMultiples An example of a k-weakly divisible set is the subset of {1, ..., N} containing the multiples of the first k primes
Erdos56.erdos_56 Suppose $A \subseteq \{1,\dots,N\}$ is such that there are no $k+1$ elements of $A$ which are relatively prime
Erdos562.erdos_562 Documented declaration: erdos_562
Erdos563.HasBalancedSubsets Documented declaration: HasBalancedSubsets
Erdos563.F Documented declaration: F
Erdos563.erdos_563 Documented declaration: erdos_563
Erdos564.erdos_564 Documented declaration: erdos_564
Erdos566.erdos_566 Let $G$ be such that any subgraph on $k$ vertices has at most $2k-3$ edges
Erdos567.Q3 $Q_3$ is the 3-dimensional hypercube graph (8 vertices, 12 edges)
Erdos567.K33 $K_{3,3}$ is the complete bipartite graph with partition sizes 3, 3 (6 vertices, 9 edges)
Erdos567.H5 $H_5$ is $C_5$ with two vertex-disjoint chords (5 vertices, 7 edges)
Erdos567.erdos_567.parts.i Erdős Problem 567 (Q3) Is $Q_3$ (the 3-dimensional hypercube) Ramsey size linear?
Erdos567.erdos_567.parts.ii Erdős Problem 567 (K33) Is $K_{3,3}$ Ramsey size linear?
Erdos567.erdos_567.parts.iii Erdős Problem 567 (H5) Is $H_5$ ($C_5$ with two vertex-disjoint chords) Ramsey size linear?
Erdos568.erdos_568 Let $G$ be a graph such that $R(G,T_n)\ll n$ for any tree $T_n$ on $n$ vertices and $R(G,K_n)\ll n^2$
Erdos569.erdos_569 Let $k\geq 1$
Erdos570.erdos_570 Let $k\geq 3$
Erdos572.erdos_572 Show that for $k\geq 3$ $$\mathrm{ex}(n;C_{2k})\gg n^{1+\frac{1}{k}}.$$ This problem is #46 in Extremal Graph Theory in the graphs problem collection
Erdos579.octahedron The octahedron $K_{2,2,2}$: the complete tripartite graph with all three parts of size $2$
Erdos579.erdos_579 Let $\delta > 0$
Erdos579.erdos_579.variants.ehss_large_delta The partial result of Erdős, Hajnal, Sós, and Szemerédi [EHSS83]: the statement of erdos_579 holds whenever the edge-density coefficient exceeds $1/8$
Erdos579.erdos_579.variants.octahedron_not_free Sanity check that the forbidden structure is non-trivial: the octahedron is of course not octahedron-free, since it contains a copy of itself
Erdos58.erdos_58 If $G$ is a graph which contains odd cycles of $\leq k$ different lengths then $\chi(G)\leq 2k+2$, with equality if and only if $G$ contains $K_{2k+2}$
Erdos581.f f m is the maximal k such that every triangle-free graph on m edges contains a bipartite subgraph with k edges
Erdos582.EdgeRamseyTriangle A graph G is edge-Ramsey for triangles (in arrow notation
Erdos582.erdos_582 Does there exist a graph $G$ which contains no $K_4$, and yet any $2$-colouring of the edges produces a monochromatic $K_3$?
Erdos583.IsPathSubgraph A subgraph H of G is a path subgraph if it is the subgraph traced out by a path in G, i.e. a walk with no repeated vertices
Erdos583.erdos_583 Every connected graph on $n$ vertices can be partitioned into at most $\lceil n/2\rceil$ edge-disjoint paths
Erdos587.MaxNotSqSum MaxNotSqSum N is the size of the largest subset A of {1,...,N} such that for all non-empty S ⊆ A, the sum ∑ n ∈ S, n is not a square
Erdos587.erdos_587.variants.nguyen_vu Nguyen and Vu proved that $|A| \ll N^{1/3} (\log N)^{O(1)}$
Erdos590.erdos_590 Let $α$ be the infinite ordinal $\omega^{\omega}$
Erdos590.erdos_590.variants.two Specker [Sp57] proved that when $α=ω^2$ any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$
Erdos590.erdos_590.variants.ge_three_false Specker [Sp57] proved that when $α=ω^n$ for $3≤ n < \omega$ then it is not the case that any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$
Erdos590.erdos_590.variants.finite_cardinal Let m be a finite cardinal $< \omega$
Erdos591.erdos_591 Let $α$ be the infinite ordinal $\omega^{\omega^2}$
Erdos592.erdos_592 Determine which countable ordinals $β$ have the property that, if $α = \omega^β$, then in any red/blue colouring of the edges of $K_α$ there is either a red $K_α$ or a blue $K_3$
Erdos593.erdos_593 Erdős Problem 593 ($500): Characterize those finite 3-uniform hypergraphs which appear in every 3-uniform hypergraph of chromatic number $> \aleph_0$
Erdos593.erdos_593.variants.obligatory_implies_two_colorable Necessary direction: every obligatory finite 3-uniform hypergraph is 2-colorable
Erdos593.erdos_593.variants.two_colorable_implies_obligatory Sufficient direction fails: it is not the case that every finite 2-colorable 3-uniform hypergraph is obligatory
Erdos593.commonPair The 3-uniform hypergraph on four vertices consisting of two triples sharing a pair, $\{0,1,2\}$ and $\{0,1,3\}$
Erdos593.erdos_593.variants.commonPair_isTwoColorable Two triples sharing a pair are 2-colorable: color the shared pair with one color and the remaining two vertices with the other
Erdos593.erdos_593.variants.common_pair_not_obligatory Two triples sharing a pair are not obligatory: by [EHR73] (see [EGH75, p
Erdos593.erdos_593.variants.graph_case_bipartite_obligatory Graph analogue — bipartite graphs are obligatory (Erdős–Galvin–Hajnal [EGH75]): For the 2-uniform (graph) case, a graph of chromatic cardinal
Erdos593.erdos_593.variants.graph_case_no_odd_cycle Graph analogue — no odd cycle is obligatory (Erdős–Galvin–Hajnal [EGH75]): For every odd
Erdos593.erdos_593.variants.uncountable_vertices_if_large_chromatic Vertices must be uncountable: Every 3-uniform hypergraph with chromatic cardinal $> \aleph_0$ must have an uncountable vertex set
Erdos593.erdos_593.variants.nonempty_edges_if_large_chromatic No hyperedges implies chromatic cardinal ≤ 1: A 3-uniform hypergraph with no edges can be properly colored with a single color, so its chromatic cardinal is at most 1
Erdos593.erdos_593.variants.obligatory_monotone Monotonicity of the obligatory property: If F₁ appears in F₂ and F₂ is obligatory, then F₁ is also obligatory
Erdos593.erdos_593.variants.empty_hypergraph_obligatory The empty hypergraph is trivially obligatory: The 3-uniform hypergraph on PEmpty (no vertices, no edges) appears in every hypergraph via the empty injection
Erdos594.erdos_594 Erdős Problem 594 (Erdős–Hajnal [ErHa66], [Er69b]): Does every graph $G$ with chromatic number $\geq \aleph_1$ contain all sufficiently large odd cycles?
Erdos594.erdos_594.variants.erdos_hajnal The earlier result of Erdős and Hajnal [ErHa66]: every graph with chromatic number $\geq \aleph_2$ contains all sufficiently large odd cycles
Erdos594.erdos_594.variants.bot_countably_colorable Sanity check for the encoding of "chromatic number
Erdos595.erdos_595 Erdős Problem 595 ($250): Is there an infinite graph $G$ which contains no $K_4$ and is not the union of countably many triangle-free graphs?
Erdos595.erdos_595.variants.folkman_finite Folkman–Nešetřil–Rödl (finite version) [Fo70, NeRo75]: For every n ≥ 1, there exists a graph G (on a finite vertex set) that contains no
Erdos595.erdos_595.variants.subgraph_of_countable_union Monotonicity: If G is a countable union of triangle-free graphs and H ≤ G (i.e., H is a subgraph of G), then H is also a countable union of triangle-free graphs
Erdos595.erdos_595.variants.triangle_free_is_union Triangle-free graphs are trivially countable unions of triangle-free graphs: if G is already triangle-free, then G = ⨆ i : ℕ, G_i where G_0 = G and G_i = ⊥ for i ≥ 1
Erdos595.erdos_595.variants.complete_nat_is_union Documented declaration: complete_nat_is_union
Erdos595.erdos_595.variants.K4_not_cliqueFree The complete graph ⊤ on Fin 4 is not $K_4$-free: ⊤ on Fin 4 equals the complete graph $K_4$, so it contains $K_4$ as a subgraph and is not $K_4$-free
Erdos595.erdos_595.variants.reformulation_edge_colouring Documented declaration: reformulation_edge_colouring
Erdos596.erdos_596 Erdős Problem 596 (Erdős–Hajnal, [Er87])
Erdos596.erdos_596.variants.exists_exceptional Erdős–Hajnal exceptional pairs exist — recorded as a known direction of erdos_596
Erdos596.erdos_596.variants.C4_free_countable_escape Every $C_4$-free graph is a countable union of trees (Erdős–Hajnal [Er87]); trees are acyclic, hence $C_6$-free, giving the countable Ramsey escape for $(C_4, C_6)$
Erdos596.erdos_596.variants.C4_C6_finite_ramsey Nešetřil–Rödl [NeRo75]: for every $n \geq 1$ there is a $C_4$-free graph whose edges cannot be $n$-coloured without a monochromatic $C_6$
Erdos596.erdos_596.variants.C4_C6_is_exceptional The pair $(C_4, C_6)$ is Erdős–Hajnal exceptional; combines C4_C6_finite_ramsey and C4_free_countable_escape
Erdos596.erdos_596.variants.original_conjecture_is_false The original Erdős–Hajnal conjecture (that no exceptional pair exists) is false — witnessed by $(C_4, C_6)$ via C4_C6_is_exceptional
Erdos596.erdos_596.variants.K4_K3_exceptional_iff Whether $(K_4, K_3)$ is Erdős–Hajnal exceptional is precisely the content of Erdős Problem 595
Erdos596.erdos_596.variants.K4_K3_finite_ramsey Folkman 1970 / Nešetřil–Rödl [NeRo75]: for every $n \geq 1$ there is a $K_4$-free graph whose edges cannot be $n$-coloured without a monochromatic triangle
Erdos596.erdos_596.test.empty_is_free The empty graph on Fin 0 is Free of any nontrivial subgraph (vacuous)
Erdos598.κ Let $\kappa = (2^{\aleph_0})^+$
Erdos598.erdos_598 Erdős Problem 598: Let $m$ be an infinite cardinal and $\kappa$ be the successor cardinal of $2^{\aleph_0}$
Erdos599.erdos_599 Erdős Problem 599 (the Erdős–Menger conjecture)
Erdos599.erdos_599.variants.aharoni_berger Menger's theorem for infinite graphs (Aharoni–Berger [AhBe09])
Erdos599.erdos_599.test.empty_A Documented declaration: empty_A
Erdos6.erdos_6 There are infinitely many $n$ such that $d_n < d_{n+1} < d_{n+2}$, where $d$ denotes the prime gap function
Erdos6.erdos_6.variants.increasing For all $m$, there are infinitely many $n$ such that $d_n < d_{n+1} < \dots < d_{n+m}$, where $d$ denotes the prime gap function
Erdos6.erdos_6.variants.decreasing For all $m$, there are infinitely many $n$ such that $d_n > d_{n+1} \dots > d_{n+m}$, where $d$ denotes the prime gap function
Erdos60.erdos_60 Does every graph on $n$ vertices with $>\mathrm{ex}(n;C_4)$ edges contain $\gg n^{1/2}$ many copies of $C_4$?
Erdos60.erdos_60.variants.he_ma_yang Documented declaration: he_ma_yang
Erdos60.erdos_60.variants.two_copies Erdős and Simonovits conjectured that at least 2 copies of $C_4$ are guaranteed
Erdos600.Erdos600Prop Documented declaration: Erdos600Prop
Erdos600.erdos_600.parts.i Let $r \geq 2$
Erdos600.erdos_600.parts.ii Let $r \geq 2$
Erdos600.erdos_600.variants.ruzsa_szemeredi_upper_bound Ruzsa and Szemerédi [RuSz78] proved that $e(n,r)=o(n^2)$ for any fixed $r$
Erdos602.IsMonochromatic A set A ⊆ α is monochromatic under a 2-colouring f : α → Fin 2 if all elements of A receive the same colour
Erdos602.HasPropertyB A family (A_i)_{i ∈ I} of subsets of α has Property B if there exists a 2-colouring f : α → Fin 2 such that no A_i is monochromatic
Erdos602.erdos_602 Does every almost-disjoint family of countably infinite sets whose pairwise intersections all have size ≠ 1 have Property B?
Erdos602.erdos_602.variants.disjoint Trivial case: pairwise disjoint families
Erdos602.erdos_602.variants.countable_index Countable index set case
Erdos602.erdos_602.variants.single_set Intersections of size ≥ 2 suffice
Erdos602.erdos_602.variants.empty_index Empty index set
Erdos602.erdos_602.variants.unique_index Unique index set
Erdos602.erdos_602.variants.two_sets Two infinite sets with pairwise intersection of size ≠ 1
Erdos602.disjoint_without_infinite_claim A natural but FALSE relaxation of erdos_602.variants.disjoint: drop the hypothesis that each A i is infinite
Erdos602.disjoint_without_infinite_claim.disproof Formal disproof of disjoint_without_infinite_claim
Erdos603.erdos_603 Let $(A_i)$ be a family of countably infinite sets such that $\lvert A_i\cap A_j\rvert \neq 2$ for all $i\neq j$
Erdos604.pinnedDistinctDistances The pinned distance function $f(n)$
Erdos604.erdos_604 Documented declaration: erdos_604
Erdos604.erdos_604.variants.sqrt_log Given $n$ distinct points $A\subset\mathbb{R}^2$, must there be a point $x\in A$ such that $$\#\{ d(x,y) : y \in A\} \gg \frac{n}{\sqrt{\log n}}?$$
Erdos604.erdos_604.variants.grid_upper_bound Documented declaration: grid_upper_bound
Erdos604.erdos_604.variants.katz_tardos Katz and Tardos [KaTa04] proved that some point of $A$ has $\gg n^{c-o(1)}$ distinct distances to the other points of $A$, where $$c=\frac{48-14e}{55-16e}=0.864137\cdots.$$
Erdos604.erdos_604.variants.average Let $d(x)$ be the number of distinct distances from $x$ to the other points of $A$
Erdos609.monochromaticGraph The monochromatic subgraph of color i under edge coloring c : Sym2 V → Fin n
Erdos609.MonochromaticHasOddCycleLe A coloring c has a monochromatic odd cycle of length at most m
Erdos609.f $f(n)$ is the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$
Erdos609.erdos_609 Let $f(n)$ be the minimal $m$ such that if the edges of $K_{2^n+1}$ are coloured with $n$ colours then there must be a monochromatic odd cycle of length at most $m$
Erdos61.IsErdosHajnalLowerBound For a graph $H$, consider all graphs $G$ that do not contain $H$ as an induced subgraph
Erdos61.erdos_61 The Erdős–Hajnal Conjecture states that there is a constant $c(H) > 0$ for each $H$ such that we can take $f(n) = n^{c(H)}$ in the above formulation
Erdos61.erdos_61.variants.erha89 Erdős and Hajnal [ErHa89] proved that we can take $f(n) = \exp(c_H \sqrt{\log n})$ for some constant $c_H > 0$ depending on $H$
Erdos61.erdos_61.variants.bnss23 Bucić, Nguyen, Scott, and Seymour [BNSS23] improved this to $f(n) = \exp(c_H \sqrt{\log n \log \log n})$ for some constant $c_H > 0$ depending on $H$
Erdos61.erdos_61.variants.p5 Documented declaration: p5
Erdos61.erdos_61.variants.c5 Documented declaration: c5
Erdos613.erdos_613 Erdős Problem 613: Let $n \geq 3$ and $G$ be a graph with $\binom{2n+1}{2} - \binom{n}{2} - 1$ edges
Erdos615.erdos_615.variants.fox_loh_zhao Documented declaration: fox_loh_zhao
Erdos615.erdos_615.variants.sudakov Documented declaration: sudakov
Erdos615.erdos_615.variants.test_bot Documented declaration: test_bot
Erdos616.erdos_616 Let $r\geq 3$
Erdos617.erdos_617 Let $r\geq 3$
Erdos617.erdos_617.variants.r_eq_3 Erdős and Gyárfás [ErGy99] proved the conjecture for $r=3$
Erdos617.erdos_617.variants.r_eq_4 Erdős and Gyárfás [ErGy99] proved the conjecture for $r=4$
Erdos617.erdos_617.variants.r2 Erdős and Gyárfás [ErGy99] showed this property fails for infinitely many $r$ if we replace $r^2+1$ by $r^2$
Erdos618.h2 Documented declaration: h2
Erdos618.erdos_618 For a triangle-free graph $G$ let $h_2(G)$ be the smallest number of edges that need to be added to $G$ so that it has diameter $2$ and is still triangle-free
Erdos619.minNewEdges For a graph
Erdos619.erdos_619.variants.add_edges_diam_three Erdős, Gyárfás and Ruszinkó [EGR98]: every connected triangle-free graph on a finite vertex set can be extended, by adding edges, to a triangle-free graph of diameter at most $3$
Erdos619.erdos_619.variants.h_three_le Erdős, Gyárfás and Ruszinkó [EGR98]: $h_3(G) \leq n$ for every connected triangle-free graph $G$ on $n$ vertices
Erdos619.erdos_619.variants.h_five_le Erdős, Gyárfás and Ruszinkó [EGR98]: $h_5(G) \leq \frac{n-1}{2}$ for every connected triangle-free graph $G$ on $n$ vertices
Erdos619.erdos_619.variants.h_three_lower Erdős, Gyárfás and Ruszinkó [EGR98]: there is a constant $c > 0$ such that for every $n$ there exist connected triangle-free graphs $G$ on $n$ vertices with $h_3(G) \geq n - c$
Erdos619.erdos_619.test.minNewEdges_eq_zero A triangle-free graph that already has diameter at most r needs no new edges
Erdos619.erdos_619.test.minNewEdges_singleton The graph with a single vertex is triangle-free and has diameter 0, so it needs no new edges to reach diameter at most 4
Erdos621.erdos_621 Documented declaration: erdos_621
Erdos623.erdos_623 Let $X$ be a set of cardinality $\aleph_\omega$ and $f$ be a function from the finite subsets of $X$ to $X$ such that $f(A)\not\in A$ for all $A$
Erdos624.ExistsEventuallySurjective The condition that an integer m ensures the existence of a function f covering Fin n for all large enough subsets Y
Erdos624.H Documented declaration: H
Erdos624.erdos_624 Documented declaration: erdos_624
Erdos627.maxChromaticCliqueRatio maxChromaticCliqueRatio n is $f(n)$, the maximum of $\chi(G)/\omega(G)$ over all graphs $G$ on $n$ vertices
Erdos627.erdos_627 Let $\omega(G)$ denote the clique number of $G$ and $\chi(G)$ the chromatic number
Erdos627.erdos_627.variants.theta Erdős [Er67c] proved that $$f(n) \asymp \frac{n}{(\log_2 n)^2}.$$
Erdos627.erdos_627.variants.lower Erdős [Er67c] proved the lower bound $$f(n) \geq \left(\frac{1}{4} + o(1)\right) \frac{n}{(\log_2 n)^2}.$$
Erdos627.erdos_627.variants.upper The method of Erdős [Er67c] gives the upper bound
Erdos627.erdos_627.variants.limit_bounds Consequently, if the limit in Erdos627.erdos_627 exists, it lies in $[1/4, 4]$ [Er67c] (with the correction of the upper constant from $1$ to $4$ noted in [AFM25])
Erdos627.erdos_627.variants.afm_upper Documented declaration: afm_upper
Erdos627.erdos_627.variants.conditional Documented declaration: conditional
Erdos627.erdos_627.variants.tutte_zykov Tutte and Zykov [Zy52] independently proved that for every $k$ there is a graph with $\omega(G) = 2$ and $\chi(G) = k$
Erdos627.erdos_627.variants.erdos_1961 Erdős [Er61d] proved that for every (large) $n$ there is a graph on $n$ vertices with $\omega(G) = 2$ and $$\chi(G) \gg \frac{n^{1/2}}{\log n},$$ whence $f(n) \gg n^{1/2}/\log n$
Erdos628.erdos_628 Let $G$ be a graph with chromatic number $k$ containing no $K_k$
Erdos628.erdos_628.variants.k_5_a_3_b_3 Erdős [Er68b] originally asked about $a=b=3$ which was proved by Brown and Jung [BrJu69] (who in fact prove that $G$ must contain two vertex disjoint odd cycles)
Erdos628.erdos_628.variants.quasi_line Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for quasi-line graphs
Erdos628.erdos_628.variants.independence_number_2 Balogh, Kostochka, Prince, and Stiebitz [BKPS09] proved the conjecture for graphs with independence number 2
Erdos633.IsCuttable A triangle is n-cuttable if it can be decomposed into n congruent triangles
Erdos633.IsCuttable.ne_zero A triangle isn't cuttable into zero triangles
Erdos633.IsCuttable.sq Every triangle is cuttable into any non-zero square number of congruent triangles
Erdos633.IsCuttable.of_isSquare Every triangle is cuttable into any non-zero square number of congruent triangles
Erdos633.isCuttable_iff_isSquare_of_linearIndependent A triangle whose side lengths and angles are integrally independent is cuttable only into a non-zero square number of congruent triangles
Erdos633.erdos_633 Which triangles can only be decomposed into a square number of congruent triangles?
Erdos633.IsSimiliCuttable A triangle is n-simili-cuttable if it can be decomposed into n similar triangles
Erdos633.IsSimiliCuttable.ne_zero A triangle isn't simili-cuttable into zero triangles
Erdos633.IsSimiliCuttable.of_ne_zero_two_three_five Every triangle is simili-cuttable into any number of similar triangles, except 0, 2, 3, 5
Erdos633.exists_isSimiliCuttable_iff_ne_zero_two_three_five There exists a triangle which isn't simili-cuttable into 0, 2, 3, 5 parts
Erdos639.erdos_639 Is it true that if the edges of $K_n$ are 2-coloured then there are at most $n^2/4$ many edges which do not occur in a monochromatic triangle?
Erdos64.erdos_64 Does every finite graph with minimum degree at least $3$ contain a cycle of length $2^k$ for some $k \geq 2$?
Erdos643.erdos_643.parts.i Documented declaration: i
Erdos643.erdos_643.parts.ii Is it true that $f(n;t)=(1+o(1))\binom{n}{t-1}$ for $t\geq 3$?
Erdos644.erdos_644.parts.i Documented declaration: i
Erdos644.erdos_644.parts.ii For every $r\geq 3$, is there $c_r>0$ such that $f(k,r)=(1+o(1))c_rk$?
Erdos645.erdos_645 If ℕ is $2$-coloured then there must exist a monochromatic three-term arithmetic progression $x,x+d,x+2d$ such that $d>x$
Erdos646.erdos_646 Let $p_1,\ldots,p_k$ be distinct primes
Erdos647.erdos_647 Let $\tau(n)$ count the number of divisors of $n$
Erdos647.erdos_647.variants.twenty_four This is true for $n = 24$
Erdos647.erdos_647.variants.lim Erdős says 'it is extremely doubtful' that there are infinitely many such $n$, and in fact suggests that $$ lim_{n\to\infty} \max_{m < n}(\tau(m) + m − n) = \infty. $$
Erdos647.erdos_647.variants.infinite Erdős says it 'seems certain' that for every $k$ there are infinitely many $n$ for which $$ \max_{n−k < m < n}(m + \tau(m)) ≤ n + 2. $$
Erdos648.g g n is the largest t such that there exist integers 2 ≤ a 0 < a 1 < ⋯ < a (t - 1) < n whose greatest prime factors are strictly decreasing
Erdos648.erdos_648 Let $g(n)$ denote the largest $t$ such that there exist integers $2\leq a_1P(a_2)>\cdots >P(a_t)$$ where $P(m)$ is the greatest prime factor of $m$
Erdos649.erdos_649 Let $P(m)$ denote the greatest prime factor of $m$
Erdos649.erdos_649.variants.no_solution_two_seven In fact, the answer to this question as written is easily seen to be no, since there are no solutions to $2^k\equiv -1\pmod{7}$, and hence this fails with $p=2$ and $q=7$
Erdos649.erdos_649.variants.tong Even with such amendments, this problem is false in a strong sense: Alan Tong has provided the following elegant elementary proof that, for any given prime
Erdos649.erdos_649.variants.tong_question Tong asks whether, for any given odd prime $q$, there are infinitely many primes $p$ such that there is no integer $n$ with $P(n)=p$ and $P(n+1)=q$
Erdos649.erdos_649.variants.sampaio Documented declaration: sampaio
Erdos649.erdos_649.variants.rmm_2020 Problem 6 in the 12th Romanian Master of Mathematics Competitions in 2020 was to prove that there exist infinitely many odd primes
Erdos65.erdos_65.parts.i Let $G$ be a graph with $n$ vertices and $kn$ edges, and $a_10$ and $k>0$
Erdos65.erdos_65.parts.ii Is the sum $\sum\frac{1}{a_i}$ minimised when $G$ is a complete bipartite graph?
Erdos650.f Documented declaration: f
Erdos650.erdos_650.variants.erdos_suranyi Erdős and Surányi [ErSu59] proved that $f(m)\geq\sqrt{m}$
Erdos650.erdos_650.variants.erdos_selfridge Erdős and Selfridge proved (see [Er78] and [Er86c]) that $f(m^2)\leq 2m$, which implies $f(m)\leq 2\lceil \sqrt{m}\rceil$ for all $m$
Erdos653.erdos_653 Documented declaration: erdos_653
Erdos655.IsValid A collection $x_1, \dots, x_n\in\mathbb{R}^2$ is valid if no circle whose centre is one of the $x_i$ contains three other points
Erdos655.erdos_655 Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points
Erdos655.erdos_655.variants.general_position Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points
Erdos659.erdos_659 Documented declaration: erdos_659
Erdos66.erdos_66 Is there and $A \subset \mathbb{N}$ is such that $$\lim_{n\to \infty}\frac{1_A\ast 1_A(n)}{\log n}$$ exists and is $\ne 0$?
Erdos660.IsPolyhedronVertices Documented declaration: IsPolyhedronVertices
Erdos660.erdos_660 Let $x_1, \ldots, x_n \in \mathbb{R}^3$ be the vertices of a convex polyhedron
Erdos660.erdos_660.variants.altman_planar For the similar problem in $\mathbb{R}^2$ there are always at least $n/2$ distances, as proved by Altman [Al63]
Erdos660.erdos_660.variants.Er75f In [Er75f] Erdős claims that Altman proved that the vertices determine $\gg n$ many distinct distances, but gives no reference
Erdos661.crossDistanceSet The set of distances $d(x, y)$ with $x \in A$ and $y \in B$
Erdos661.minimalCrossDistances The minimal number of distinct distances $d(x_i, y_j)$ over all choices of $2n$ distinct points $x_1, \ldots, x_n, y_1, \ldots, y_n$ in X
Erdos661.erdos_661 Are there, for all large
Erdos661.erdos_661.variants.grid_upper_bound Take $2n$ points of the smallest square grid with at least $2n$ points; by [Er46] they determine $O(n / \sqrt{\log n})$ distinct distances
Erdos661.erdos_661.variants.le_minimalDistinctDistances Every split of a set of $2n$ points into two halves of $n$ points gives a configuration for $F(n)$, and the distances between the halves are among the distances of the whole set
Erdos661.erdos_661.variants.little_o_distinct_distances Documented declaration: little_o_distinct_distances
Erdos661.erdos_661.variants.lenz Documented declaration: lenz
Erdos664.erdos_664.variants.block_design In [Er81] the condition
Erdos665.erdos_665 Documented declaration: erdos_665
Erdos666.erdos_666 Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges)
Erdos667.LocallyDense Documented declaration: LocallyDense
Erdos667.H H p q n is H(n) = H(n; p, q): the largest m such that every (p, q)-locally dense graph on n vertices must contain a complete graph on m vertices Kₘ
Erdos667.c The growth exponent c(p, q) = liminf_{n → ∞} (log H(n)) / (log n)
Erdos667.erdos_667 Let $p, q \geq 1$ be fixed integers
Erdos667.erdos_667.variants.monotoneOn Documented declaration: monotoneOn
Erdos667.erdos_667.variants.endpoint The right endpoint: when $q = \binom{p-1}{2} + 1$ one has $c(p, q) = 1$ (stated as "easy to see" on erdosproblems.com/667)
Erdos667.erdos_667.variants.ramsey_bounds The case $q = 1$ is the classical Ramsey problem, giving the bounds $\tfrac{1}{p-1} \leq c(p, 1) \leq \tfrac{2}{p+1}$ for $p \geq 3$
Erdos67.erdos_67 Documented declaration: erdos_67
Erdos67.erdos_67.variants.complex Documented declaration: complex
Erdos670.HasSeparatedDistances A finite set
Erdos670.erdos_670 Let $A\subseteq \mathbb{R}^d$ be a set of $n$ points such that all pairwise distances differ by at least $1$
Erdos670.erdos_670.variants.dim_one Documented declaration: dim_one
Erdos670.erdos_670.variants.choose_two Documented declaration: choose_two
Erdos670.erdos_670.variants.uniform_in_dim The version of the problem where the $o(1)$ term must be uniform in the dimension $d$ has a negative answer
Erdos672.Erdos672With Erdős problem 672 conjectures that the below holds for any $k ≥ 4$ and $l > 1$
Erdos672.erdos_672 Can the product of an arithmetic progression of positive integers $n, n + d, ..., n + (k - 1)d$ of length $k ≥ 4$, with $(n, d) = 1$, be a perfect power?
Erdos672.erdos_672.variants.euler According to https://www.erdosproblems.com/672, Euler proved this
Erdos672.erdos_672.variants.oblath According to https://www.erdosproblems.com/672, Obláth proved this
Erdos673.G If $1=d_1<\cdots <d_{\tau(n)}=n$ are the divisors of $n$ (here Nat.nth (· ∣ n) i is $d_{i+1}$), then $$G(n) = \sum_{1\leq i<\tau(n)}\frac{d_i}{d_{i+1}}.$$
Erdos673.erdos_673.variants.tao Tao observed that, for any divisor $m\mid n$ with $m>1$, $\frac{\tau(n/m)}{m} \leq G(n) \leq \tau(n)$
Erdos674.solutionSet The set of integer solutions to $x^x y^y = z^z$ with $x, y, z > 1$
Erdos674.erdos_674 Are there any integer solutions to $x^xy^y=z^z$ with $x,y,z>1$?
Erdos674.erdos_674.variants.infinite There are in fact infinitely many integer solutions to $x^xy^y=z^z$ with $x,y,z>1$
Erdos677.lcmInterval_eq_example1 Erdős expected very few solutions for $M(n, k) = M(m, l)$, where $m \geq n + k$ and $l > 1$
Erdos677.erdos_677 Denote by $M(n, k)$ the least common multiple of the finite set $\{n+1, \dotsc, n+k\}$
Erdos678.lcmInterval_lt_example1 The referee of [Er79] found the example $M(96, 7) > M(104, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$
Erdos678.lcmInterval_lt_example2 The referee of [Er79] found the example $M(132, 7) > M(139, 8)$, showing that there are cases where $M(n, k) > M(m, k + 1)$ with $m \geq n + k$
Erdos678.lcmInterval_lt_example3 Cambie [Ca24] found the example $M(52, 7) > M(62, 8)$
Erdos678.lcmInterval_lt_example4 Cambie [Ca24] found the example $M(36, 8) > M(48, 9)$
Erdos678.erdos_678 Write $M(n, k)$ be the least common multiple of $\{n+1, \dotsc, n+k\}$
Erdos678.erdos_678.variants.infinitely_many_triples Documented declaration: infinitely_many_triples
Erdos678.erdos_678.variants.not_infinitely_many_pairs For a fixed sufficiently large $k$ only finitely many pairs occur: $M(m, k + 1) \geq m + 1$ bounds $m$ by $M(n, k)$, and for large $n$ the inequality reverses
Erdos68.erdos_68 Is $$\sum_{n=2}^\infty \frac{1}{n!-1}$$ irrational?
Erdos68.sum_factorial_inv_eq_geometric $$\sum_{n=2}^\infty \frac{1}{n!-1} = \sum_{n=2}^\infty \sum_{k=1}^\infty \frac{1}{(n!)^k}$$
Erdos680.erdos_680.parts.i Is it true that, for all sufficiently large $n$, there exists some $k$ such that $$ p(n+k)>k^2+1, $$ where $p(m)$ denotes the least prime factor of $m$?
Erdos680.erdos_680.parts.ii Can one prove this is false if we replace $k^2+1$ by $e^{(1+\epsilon)\sqrt{k}}+C_\epsilon$, for all $\epsilon>0$, where $C_\epsilon>0$ is some constant?
Erdos681.IsLPF IsLPF p m says that p is the least prime factor of m
Erdos681.erdos_681 Erdős problem 681
Erdos682.p The nth prime, indexed from zero: p 0 = 2, p 1 = 3,
Erdos682.erdos_682 Is it true that for almost all $n$ there exists some $m\in (p_n,p_{n+1})$ such that $$p(m) \geq p_{n+1}-p_n,$$ where $p(m)$ denotes the least prime factor of $m$?
Erdos682.erdos_682.variants.exceptional_count Gafni and Tao [GaTa25, Theorem 1.1] proved that the number of exceptional gaps $(p_n,p_{n+1})$ with $p_n\in [X,2X]$ is $$\ll \frac{X}{(\log X)^2}.$$
Erdos682.erdos_682.variants.dickson_counterexample Erdős first thought this should be true for all large
Erdos683.P Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$
Erdos683.erdos_683 Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$
Erdos683.erdos_683.variant.sylvester_schur Sylvester and Schur [Er34] proved that $P(n, k) > k$ for $k \le n/2$
Erdos683.erdos_683.variant.erdos_log Erdős [Er55d] improved this to $P(n, k) \gg \min(n - k + 1, k \log k)$ for $k \le n/2$
Erdos683.erdos_683.variant.exp_sqrt Standard heuristics suggest that $P(n, k) > \min(n - k + 1, e^{c\sqrt{k}})$ for some constant $c > 0$
Erdos686.erdos_686 Can every integer $N≥2$ be written as $$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?
Erdos686.erdos_686.variants.square Can every square $N≥2$ be written as $$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?
Erdos686.erdos_686.variants.four Can $4$ be written as $$4=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?
Erdos686.erdos_686.variants.four_two The number $4$ cannot be written as $$4=\frac{\prod_{1\leq i\leq 2}(m+i)}{\prod_{1\leq i\leq 2}(n+i)}$$ for $m≥n+2$!
Erdos686.erdos_686.variants.four_three The number $4$ cannot be written as $$4=\frac{\prod_{1\leq i\leq 3}(m+i)}{\prod_{1\leq i\leq 3}(n+i)}$$ for $m≥n+3$!
Erdos686.erdos_686.variants.nine Can $9$ be written as $$9=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?
Erdos686.erdos_686.variants.twenty_five Can $25$ be written as $$25=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?
Erdos686.erdos_686.variants.non_square Can every non-square $N≥2$ be written as $$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$ for some $k≥2$ and $m≥n+k$?
Erdos688.Erdos688Prop Define
Erdos688.erdos_688.parts.i Estimate $\epsilon_n$
Erdos688.erdos_688.parts.ii In particular, is it true that $\epsilon_n = o(1)$?
Erdos688.erdos_688.variants.lglglg_over_lglg_is_big_o Erdős claims in [Er80] (p
Erdos689.erdos_689 Let n be sufficiently large
Erdos69.erdos_69 Is $$ \sum_{n\geq 2}\frac{\omega(n)}{2^n} $$ irrational?
Erdos69.erdos_69.variants.specialisation_of_erdos_257 Tao observed that erdos_69 is a special case of erdos_257, since $$ \sum_{n\geq 2}\frac{\omega(n)}{2^n} = \sum_p \frac{1}{2^p - 1}. $$
Erdos690.kthPrimeFactorSet The set of positive integers whose k-th smallest prime factor is p: if $p_1<p_2<\cdots$ are the primes dividing $n$ then $p_k=p$
Erdos690.IsUnimodalOnPrimes A function on the primes is unimodular if it first increases in $p$ until its maximum then decreases
Erdos690.erdos_690.variants.large_k Is $d_k(p)$ unimodular for any $k\geq 21$?
Erdos692.exactlyOneDivisorIn The set of integers with exactly one divisor in the open interval $(n, m)$
Erdos692.IsDelta₁ $\delta_1(n,m)$ is the density of the set of integers with exactly one divisor in $(n,m)$
Erdos692.localMaxima The set of local maxima of the sequence f on (a, ∞)
Erdos692.erdos_692.parts.ii Let $\delta_1(n,m)$ be the density of the set of integers with exactly one divisor in $(n,m)$
Erdos692.erdos_692.variants.erdos_upper_bound Erdős proved that $$\delta_1(n,m) \ll \frac{1}{(\log n)^c}$$ for all $m$, for some constant $c>0$
Erdos692.erdos_692.variants.cambie_three Cambie has calculated that unimodularity fails even for $n=2$ and $n=3$
Erdos692.erdos_692.variants.cambie_local_maxima Cambie [Ca25] has shown that, for fixed $n$, the sequence $\delta_1(n,m)$ has superpolynomially many local maxima $m$
Erdos694.erdos_694 Let $f_\max(n)$ be the largest $m$ such that $\phi(m) = n$, and $f_\min(n)$ be the smallest such $m$, where $\phi$ is Euler's totient function
Erdos694.erdos_694.variants.carmichael Carmichael has asked whether there is an integer $n$ for which $\phi(m) = n$ has exactly one solution, that is $\frac{f_\max(n)}{f_\min(n)} = 1$
Erdos694.erdos_694.variants.inf_unique Erdős has proved that if there exists an integer $n$ for which $\phi(m) = n$ has exactly one solution, then there must be infinitely many such $n$
Erdos695.erdos_695 Let $q_1 < q_2 < \cdots$ be a sequence of primes such that $q_{i + 1} \equiv 1 \pmod{q_i}$
Erdos695.erdos_695.variants.upperBound Is there a sequence of primes $q_1 < q_2 < \cdots$ such that $q_{i + 1} \equiv 1 \pmod{q_i}$ and $$ q(k) \leq \exp(k (\log k)^{1 + o(1)})? $$
Erdos697.density_exists For each $m$ and $\alpha$, the density of the set of integers which are divisible by some $d \equiv 1 \pmod{m}$ with $1 < d < \exp (m ^ \alpha)$ exists
Erdos697.δ For each $m$ and $\alpha$, $\delta (m, \alpha)$ is the density of the set of integers which are divisible by some $d \equiv 1 \pmod{m}$ with $1 < d < exp (m ^ \alpha)$ exists
Erdos697.erdos_697.variants.delta_lt $\delta < \frac{m ^ \alpha + 1}{m}$
Erdos697.erdos_697.parts.i Let $\beta = \frac{1}{\log 2}$
Erdos697.erdos_697.parts.ii $lim_{m\rightarrow\infty} \delta (m, \alpha) = 1$ if $\beta < \alpha$
Erdos698.erdos_698.variants.erdos_szekeres Documented declaration: erdos_szekeres
Erdos698.erdos_698.variants.erdos_szekeres_sharp This inequality is sharp for $i=1$, $j=p$, and $n=2p$
Erdos698.erdos_698.variants.bergman Documented declaration: bergman
Erdos699.sylvester_schur Sylvester and Schur: for $1 \le i \le n/2$ there is a prime $p > i$ dividing n.choose i
Erdos699.erdos_699 Erdős Problem 699
Erdos699.erdos_szekeres_strengthening Erdős and Szekeres conjectured that, apart from a finite exceptional set of triples (n, i, j), one can always take p > i in the prime divisor statement
Erdos7.erdos_7 Is there a covering system all of whose moduli are odd (and greater than 1)?
Erdos70.OrdinalCardinalRamsey3 OrdinalCardinalRamsey3 α β c asserts the 3-uniform ordinal Ramsey property $\alpha \to (\beta, c)^3_2$
Erdos70.RealCardinalRamsey3 RealCardinalRamsey3 β c asserts the 3-uniform partition relation
Erdos70.erdos_70 Erdős Problem 70: Let $\mathfrak{c}$ be the order type of the real numbers, let $\beta$ be a countable ordinal, and let $2 \le n < \omega$
Erdos70.erdos_70.variants.erdos_rado Erdős–Rado partial result: $\mathfrak{c} \to (\omega + n, 4)^3_2$ for any $2 \le n < \omega$, where $\mathfrak{c}$ is the order type of the real numbers
Erdos70.erdos_70.variants.omega_times_two_four First open case beyond Erdős–Rado: $\mathfrak{c} \to (\omega \cdot 2, 4)^3_2$, where $\mathfrak{c}$ is the order type of the real numbers
Erdos70.erdos_70.variants.omega_one The relation at $\omega_1$: $\mathfrak{c} \to (\omega_1, n)^3_2$ for finite $n \ge 2$, where $\omega_1 = \aleph_1$ is the first uncountable ordinal
Erdos70.erdos_70.variants.ordinalCardinalRamsey3_mono Monotonicity of OrdinalCardinalRamsey3: If OrdinalCardinalRamsey3 α β c holds and $\beta' \le \beta$, $c' \le c$, then OrdinalCardinalRamsey3 α β' c' also holds
Erdos700.f f n = min_{1 < k ≤ n/2} gcd(n, C(n,k))
Erdos700.P P n is the largest prime factor of n (and 0 if n ≤ 1)
Erdos700.Q $Q(n)$ is the largest prime-power divisor of $n$: the maximum of $p^{v_p(n)}$ over prime divisors $p$ of $n$
Erdos700.fSet The set whose infimum defines f
Erdos700.f_eq f n unfolds to the infimum of fSet n
Erdos700.f_mem Each gcd(n, C(n,k)) with 1 < k ≤ n/2 belongs to fSet n
Erdos700.f_le f n is a lower bound: f n ≤ gcd(n, C(n,k)) for every 1 < k ≤ n/2
Erdos700.erdos_700.parts.i Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$ and let $P(n)$ be the largest prime dividing $n$
Erdos700.erdos_700.variants.prime_power_divisor Characterise those composite $n$ such that $f(n) = n/Q(n)$, where $Q(n)$ is the largest prime-power divisor of $n$
Erdos700.prime_power_divisor_separation At $n = 12$, $f(n) = 3 = n/Q(n)$, whereas $n/P(n) = 4$
Erdos700.erdos_700.parts.ii Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$
Erdos700.erdos_700.parts.iii Let $f(n) = \min_{1 < k \le n/2} \gcd(n, \binom{n}{k})$
Erdos700.prime_dvd_of_not_dvd_choose Lucas (one step): for prime P ∣ n, if P ∤ C(n,k) then P ∣ k
Erdos700.erdos_700.variants.prime_pow f(p^a) = p for a prime p and a ≥ 2 (recorded by Erdős–Szekeres [ErSz78])
Erdos700.erdos_700.variants.prime_mul f(pq) = p for distinct primes p < q (recorded by Erdős–Szekeres [ErSz78]); in particular pq is always a "hit" (f(pq) = pq / P(pq) = p)
Erdos701.erdos_701 Let $\mathcal{F}$ be a family of sets closed under taking subsets (i.e. if $B\subseteq A\in\mathcal{F}$ then $B\in \mathcal{F}$)
Erdos703.T T n r is maximal such that there exists a family
Erdos703.erdos_703.variants.zero It is trivial that $T(n,0)=2^{n-1}$
Erdos703.erdos_703.variants.frankl_furedi Frankl and Füredi [FrFu84b] proved that, for fixed
Erdos707.erdos_707 Erdős Problem 707: It is false that any finite Sidon set can be embedded in a perfect different set modulo some $n$
Erdos707.erdos_707.variants.prime_power It is false that any finite Sidon set can be embedded in a perfect difference set modulo p^2 + p + 1 for some prime power p
Erdos707.erdos_707.variants.prime It is false that any finite Sidon set can be embedded in a perfect difference set modulo p^2 + p + 1 for some prime p
Erdos707.erdos_707.variants.counterexample_prime Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture, proving that $\{1,2,4,8\}$ cannot be extended to a perfect difference set modulo $p^2+p+1$ for any prime $p$
Erdos707.erdos_707.variants.counterexample_mian_chowla Alexeev and Mixon [arxiv/2510.19804] have disproved this conjecture, showing that $\{1, 2, 4, 8, 13\}$ cannot be extended to any perfect difference set modulo a positive integer
Erdos707.erdos_707.variants.counterexample_hall This conjecture was actually first disproved by Hall in 1947 [Ha47], long before Erdős asked this question
Erdos707.erdos_707.variants.perfect_difference_set_size_bound A perfect difference set modulo n must have size ≤ √n + 1
Erdos707.erdos_707.variants.singer_construction The Singer construction gives perfect difference sets for n = p^2 + p + 1 where p is a prime power
Erdos707.erdos_707.variants.example_sidon_set The set {1, 2, 4} is a Sidon set
Erdos707.erdos_707.variants.example_embedding The set {1, 2, 4} can be embedded in a perfect difference set modulo 7
Erdos707.erdos_707.variants.small_sidon_sets For small Sidon sets, we can check the conjecture directly
Erdos71.erdos_71 Is it true that for every infinite arithmetic progression
Erdos712.erdos_712 Documented declaration: erdos_712
Erdos713.erdos_713.parts.i Is it true that, for every bipartite graph
Erdos713.erdos_713.parts.ii Must $\alpha$ be rational?
Erdos714.erdos_714 Is it true that $$\mathrm{ex}(n; K_{r,r}) \gg n^{2-1/r}?$$
Erdos716.erdos_716 Let $\mathcal{F}$ be the family of all $3$-uniform hypergraphs with $6$ vertices and $3$ $3$-edges
Erdos719.erdos_719 Documented declaration: erdos_719
Erdos720.sizeRamseyNumber The size Ramsey number $\hat{R}(G)$ of G: the least number of edges of a graph which is Ramsey for G
Erdos722.erdos_722 Let $k>r$ and $n$ be sufficiently large in terms of $k$ and $r$
Erdos723.erdos_723 If there is a finite projective plane of order $n$ then must $n$ be a prime power?
Erdos723.erdos_723.variants.prime_power_is_projplane_order These always exist if $n$ is a prime power
Erdos723.erdos_723.variants.leq_11 This conjecture has been proved for $n \leq 11$
Erdos723.erdos_723.variants.eq_12 It is open whether there exists a projective plane of order 12
Erdos723.erdos_723.variants.bruck_ryser Bruck and Ryser have proved that if $n \equiv 1 (\mod 4)$ or $n \equiv 2 (\mod 4)$ then $n$ must be the sum of two squares
Erdos726.erdos_726 As $n\to \infty$ ranges over integers $\sum_{p\leq n}1_{n\in (p/2,p)\pmod{p}}\frac{1}{p}\sim \frac{\log\log n}{2}$?
Erdos726.erdos_726.variants.mertens_estimate The classical estimate of Mertens states that $\sum_{p\leq n}\frac{1}{p}\sim \log\log n$
Erdos727.erdos_727 Let $k ≥ 2$
Erdos727.erdos_727.variants.k_2 It is open even for $k = 2$
Erdos727.erdos_727.variants.k_1 Balakran proved this holds for $k = 1$
Erdos727.erdos_727.variants.k_1_2 Erdős, Graham, Ruzsa, and Straus observe that the method of Balakran can be further used to prove that there are infinitely many $n$ such that $(n+k)!(n+1)!∣(2n)!$
Erdos728.erdos_728 Let $\varepsilon$ be sufficiently small and $C, C' > 0$
Erdos729.erdos_729 Let $C>0$ be a constant
Erdos730.erdos_730.variants.explicit_pairs For example, $(87,88)$ and $(607,608)$ are such pairs
Erdos730.erdos_730.variants.delta_ne_one There are examples where $(n, m) ∈ S$ with $m ≠ n + 1$
Erdos732.erdos_732.parts.i Call a sequence
Erdos732.erdos_732.parts.ii Is there some constant $c>0$ such that for all large $n$ there are at least $\exp(c n^{1/2}\log n)$ block-compatible sequences?
Erdos734.erdos_734 Find, for all large
Erdos74.SimpleGraph.edgeDistancesToBipartite For a given subgraph A, this is the set of all numbers k such that A can be made bipartite by deleting k edges
Erdos74.SimpleGraph.edgeDistancesToBipartite_nonempty The set of edge distances to a bipartite graph is always non-empty because deleting all edges from a graph makes it bipartite
Erdos74.SimpleGraph.minEdgeDistToBipartite The minimum number of edges that must be deleted from a subgraph A to make it bipartite
Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite For a graph G and a number n, this is the set of minEdgeDistToBipartite A for all induced subgraphs A of G on n vertices
Erdos74.SimpleGraph.subgraphEdgeDistsToBipartite_bddAbove The set of minimum edge distances to bipartite for subgraphs of size n is bounded above
Erdos74.SimpleGraph.maxSubgraphEdgeDistToBipartite For a given graph $G$ and size $n$, this defines the smallest number $k$ such that any subgraph of $G$ on $n$ vertices can be made bipartite by deleting at most $k$ edges
Erdos74.erdos_74.variants.sqrt Is there a graph of infinite chromatic number such that every finite subgraph on $n$ vertices can be made bipartite by deleting at most $\sqrt{n}$ edges?
Erdos740.NoShortOddCycle A graph avoids odd cycles of length $\leq r$ if it contains no odd cycles of length at most $r$
Erdos740.erdos_740 Let $\mathfrak{m}$ be an infinite cardinal and $G$ be a graph with chromatic number $\mathfrak{m}$
Erdos741.erdos_741.variants.lower Let $A\subseteq \mathbb{N}$ be such that $A+A$ has positive lower density
Erdos742.IsDiameter2Critical A graph is diameter-2-critical if it has diameter $2$ and removing any edge increases the diameter beyond $2$
Erdos742.erdos_742 Murty-Simon Conjecture Let $G$ be a graph on $n$ vertices with diameter $2$ such that deleting any edge increases the diameter
Erdos742.complete_bipartite_edge_count The complete bipartite graph $K_{a, b}$ has exactly $a \cdot b$ edges
Erdos742.variants.plesnik_bound Plesník [Pl75] proved the bound $|E(G)| < 3n(n-1)/8$ for any diameter-$2$-critical graph on $n$ vertices
Erdos742.variants.fan_bound Fan [Fa87] verified the Murty-Simon conjecture for all $n \leq 24$ and for $n = 26$
Erdos747.erdos_747 How large should $\ell(n)$ be such that, almost surely, a random $3$-uniform hypergraph on $3n$ vertices with $\ell(n)$ edges must contain $n$ vertex-disjoint edges?
Erdos748.f f n counts the number of sum-free $A\subseteq \{1,\ldots,n\}$, i.e. $A$ contains no solutions to $a=b+c$ with $a,b,c\in A$
Erdos748.erdos_748.variants.lower_bound It is trivial to see that $f(n) \geq 2^{\frac{n}{2}}$, considering all subsets of $(n/2,n]$
Erdos748.erdos_748.variants.green_sapozhenko Green [Gr04] and Sapozhenko [Sa03] proved that $f(n) \ll 2^{n/2}$
Erdos748.erdos_748.variants.asymptotic In fact, both Green [Gr04] and Sapozhenko [Sa03] prove the stronger asymptotic $f(n) \sim c_n 2^{n/2}$, where $c_n$ takes on one of two values depending on the parity of $n$
Erdos749.erdos_749 Let $\epsilon>0$
Erdos75.erdos_75 Documented declaration: erdos_75
Erdos750.MycVerts Vertices of the generalised Mycielskian $M_s(G)$: $s$ copies of the vertices of $G$, one per level, and a single apex
Erdos750.MycAdj Adjacency of the generalised Mycielskian, before SimpleGraph.fromRel packages it as a graph
Erdos750.genMyc The generalised Mycielskian $M_s(G)$
Erdos750.genMyc_apex_adj_top The apex of $M_s(G)$ is adjacent to every vertex on the top level
Erdos750.genMyc_adj_level_zero Two level-0 vertices of $M_s(G)$ are adjacent exactly when the underlying vertices are
Erdos750.IsRecursivelyBuiltMr G belongs to the class $M_r$, meaning it is built from $K_2$ by $r - 2$ generalised Mycielskians
Erdos750.erdos_750.variants.stiebitz Stiebitz's theorem [St85]: every graph in $M_r$ has chromatic number at least $r$
Erdos750.erdos_750.variants.epsilon In [Er69b] Erdős conjectures this for $f(m)=\epsilon m$ for any fixed $\epsilon>0$
Erdos750.erdos_750.variants.c_gt_quarter In [ErHa67b] Erdős and Hajnal prove this for $f(m)\geq cm$ for all $c>1/4$
Erdos751.erdos_751.parts.i Let $G$ be a graph with chromatic number $\chi(G)=4$
Erdos751.erdos_751.parts.ii Let $G$ be a graph with chromatic number $\chi(G)=4$
Erdos751.erdos_751.variants.bondy_vince Bondy and Vince [BoVi98] proved that every graph with minimum degree at least $3$ has two cycles whose lengths differ by at most $2$
Erdos753.IsKChoosable Documented declaration: IsKChoosable
Erdos753.listChromaticNumber The list chromatic number $\chi_L(G)$, defined to be the minimal $k$ such that $G$ is $k$-choosable
Erdos753.erdos_753 The list chromatic number
Erdos753.erdos_753.variants.alon Alon [Al92] proved that, for every $n$, there exists a graph $G$ on $n$ vertices such that $$\chi_L(G)+\chi_L(G^c)\ll (n\log n)^{1/2},$$ where the implied constant is absolute
Erdos755.IsEquilateralTriangle A three-point set whose pairwise distances are all equal to side
Erdos755.IsUnitEquilateralTriangle A unit equilateral triangle in Euclidean d-space
Erdos755.IsAnySizeEquilateralTriangle An equilateral triangle of any positive side length in Euclidean d-space
Erdos755.unitEquilateralTriangleCount Number of unit equilateral triangles spanned by a finite point set
Erdos755.anySizeEquilateralTriangleCount Number of equilateral triangles of any positive side length spanned by a finite point set
Erdos755.TUnit Maximum number of unit equilateral triangles spanned by $n$ points in $\mathbb{R}^d$
Erdos755.TAnySize Maximum number of equilateral triangles of any size spanned by $n$ points in $\mathbb{R}^d$
Erdos755.erdos_755.variants.any_size_cdl Clemen, Dumitrescu, and Liu [CDL25b] proved the stronger version where equilateral triangles of all positive side lengths are counted
Erdos756.richDistances The distances determined by A which occur for at least k many pairs of points of A
Erdos756.maxRichDistances The largest number of distinct distances that a set of n points in $\mathbb{R}^2$ can determine, each of which occurs for more than n many pairs of points of the set
Erdos756.erdos_756 Let $A\subset \mathbb{R}^2$ be a set of $n$ points
Erdos756.erdos_756.variants.bhowmick Bhowmick [Bh24] constructs a set of $n$ points in $\mathbb{R}^2$ such that $\lfloor\frac{n}{4}\rfloor$ distances occur at least $n+1$ times
Erdos756.erdos_756.variants.bhowmick_general More generally, they construct, for any positive integer $m$ and large $n$, a set of $n$ points such that $\lfloor \frac{n}{2(m+1)}\rfloor$ distances occur at least $n+m$ times
Erdos757.IsAdmissible Documented declaration: IsAdmissible
Erdos757.erdos_757 What is the supremum of the set of admissible numbers?
Erdos757.erdos_757.variants.lowerBound The supremum is strictly larger than 1 / 2, which is proved in [GyLe95]
Erdos757.erdos_757.variants.upperBound In [GyLe95], the authors also prove that the supremum is smaller than 3 / 5
Erdos760.erdos_760 Documented declaration: erdos_760
Erdos760.erdos_760.variants.erdos_gimbel A problem of Erdős and Gimbel, who proved that there must exist a subgraph $H$ with $$ \zeta(H) \gg \left(\frac{m}{\log m}\right)^{1/2}. $$
Erdos762.erdos_762 Documented declaration: erdos_762
Erdos762.erdos_762.variants.bounded_clique_number Documented declaration: bounded_clique_number
Erdos762.erdos_762.variants.steiner This has been disproved by Steiner [St24b], who constructed a graph $G$ with $\omega(G)=4$, $\zeta(G)=4$, and $\chi(G)=7$
Erdos763.erdos_763.variants.erdos_fuchs Erdős and Fuchs [ErFu56] proved that $\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)$ is impossible
Erdos763.erdos_763.variants.montgomery_vaughan Jurkat (unpublished) and Montgomery and Vaughan [MoVa90] proved that even $\sum_{n\leq N} 1_A\ast 1_A(n) = cN+o(N^{1/4})$ is impossible
Erdos764.tripleRep The number of representations of n as an ordered sum of three elements of A, $1_A\ast 1_A\ast 1_A(n)$
Erdos764.erdos_764.variants.vaughan Vaughan [Va72] showed that $\sum_{n\leq N} 1_A\ast 1_A\ast 1_A(n) = cN+o\left(\frac{N^{1/4}}{(\log N)^{1/2}}\right)$ is impossible
Erdos765.erdos_765.variants.second_term Documented declaration: second_term
Erdos767.HasCycleWithIncidentChords G contains a cycle with k chords incident to a vertex on the cycle
Erdos767.g g k n is the maximal number of edges possible on a graph with n vertices which does not contain a cycle with k chords incident to a vertex on the cycle
Erdos767.erdos_767.variants.czipszer Czipszer proved that $g_k(n)\leq (k+1)n$
Erdos767.erdos_767.variants.posa Pósa proved that $g_1(n)=2n-4$ for $n\geq 4$
Erdos769.Cube An axis-parallel homothetic copy of the standard n-cube
Erdos769.Cube.Mem Membership in the half-open cube ∏ j, [lower j, lower j + side)
Erdos769.InUnit Membership in the half-open unit cube [0,1)^n
Erdos769.Cube.InsideUnit A cube is nondegenerate and contained in the closed unit cube
Erdos769.IsTiling Documented declaration: IsTiling
Erdos769.Admissible Exactly k homothetic n-cubes can tile the unit n-cube
Erdos769.IsCutoff c is the minimal cutoff c(n): every k ≥ c is admissible, and (when c > 0) c - 1 is not
Erdos769.Erdos769LowerBound The highlighted conjecture c(n) ≫ n^n: there is an absolute positive constant A/B with c(n) ≥ (A/B) n^n for all sufficiently large n
Erdos769.erdos_769 Let $c(n)$ be minimal such that if $k\geq c(n)$ then the $n$-dimensional unit cube can be decomposed into $k$ homothetic $n$-dimensional cubes
Erdos769.erdos_769.variants.growth_rate What refuting $c(n)\gg n^n$ leaves open: give good bounds for $c(n)$
Erdos77.erdos_77 Documented declaration: erdos_77
Erdos770.h Let $h n$ be the minimal number such that $2 ^ n - 1, \dots, h(n) ^ n - 1$ are collectively coprime
Erdos770.Nat.Prime.h_eq_add_one n + 1 is prime iff h n = n + 1
Erdos770.erdos_770.variants.odd_h_unbounded For odd n, the values of h n form an unbounded set
Erdos770.erdos_770.parts.i For every prime p, does the density of integers with h n = p exist?
Erdos770.erdos_770.parts.ii Does liminf h n = ∞?
Erdos770.erdos_770.parts.iii Is it true that if p is the greatest prime such that p - 1 ∣ n and p > n ^ ε, then h n = p?
Erdos770.erdos_770.variants.three It is probably true that h n = 3 for infinitely many n
Erdos772.H $H_k(n)$ is the maximal $r$ such that if $A\subset\mathbb{N}$ has $\lvert A\rvert=n$ and $\| 1_A\ast 1_A\|_\infty \leq k$ then $A$ contains a Sidon set of size at least $r$
Erdos772.erdos_772.variants.alon_erdos Alon and Erdős [AlEr85] proved that $H_k(n) \gg_k n^{2/3}$
Erdos772.erdos_772.variants.upper_bound Erdős [Er84d] proved that $H_k(n) \ll n^{2/3}$, where the implied constant is absolute
Erdos773.erdos_773 What is the size of the largest Sidon subset $A\subseteq\{1,2^2,\ldots,N^2\}$?
Set.IsDissociated We call $A\subset \mathbb{N}$ dissociated if $\sum_{n\in X}n\neq \sum_{m\in Y}m$ for all finite $X,Y\subset A$ with $X\neq Y$
Set.IsProportionatelyDissociated We call $A$ proportionately dissociated if every finite $B\subset A$ contains a dissociated set of size $\gg \lvert B\rvert$
Erdos774.erdos_774 Is every proportionately dissociated (infinite) set the union of a finite number of dissociated sets?
Erdos775.erdos_775 Is there a $3$-uniform hypergraph on $n$ vertices which contains at least $n-O(1)$ different sizes of cliques (maximal complete subgraphs)?
Erdos775.erdos_775.variants.gao Documented declaration: gao
Erdos775.erdos_775.variants.spencer For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least $n-\log_2n+O(1)$ different sizes
Erdos775.erdos_775.variants.moon_moser For graphs, Spencer [Sp71] constructed a graph which contains cliques of at least $n-\log_2n+O(1)$ different sizes, which Moon and Moser [MoMo65] showed to be best possible
Erdos779.erdos_779 A Conjecture of Marian Deaconescu, see p.120 in https://doi.org/10.2307/2975810 [Needed to index shift in order to avoid trivial case
Erdos78.explicitGraph The graph on Fin n described by an adjacency oracle adj
Erdos78.stronglyExplicitGraph The graph on Fin n described by an adjacency oracle adj which receives n in binary
Erdos78.NoHomogeneousSet The graph G has no clique and no independent set with m vertices
Erdos78.erdos_78 Let $R(k)$ be the Ramsey number for $K_k$
Erdos78.erdos_78.variants.nonconstructive Erdős [Er47] gave a simple probabilistic, non-constructive proof that $R(k) > C^k$ for some constant $C > 1$
Erdos78.erdos_78.variants.little_o_sqrt Erdős also asked for an explicit construction of graphs on $n$ vertices whose largest clique and independent set have size $o(n^{1/2})$
Erdos78.erdos_78.variants.cohen Cohen [Co15] constructed explicit graphs on $n$ vertices with no clique and no independent set of size $2^{(\log \log n)^C}$, for some constant $C > 0$
Erdos78.erdos_78.variants.li Li [Li23b] improved this to explicit graphs on $n$ vertices with no clique and no independent set of size $(\log n)^C$, for some constant $C > 0$
Erdos78.erdos_78.variants.strongly_explicit Documented declaration: strongly_explicit
Erdos78.erdos_78.variants.li_strongly_explicit Documented declaration: li_strongly_explicit
Erdos780.erdos_780.variants.sharp Documented declaration: sharp
Erdos781.IsDescendingWave A $k$-term descending wave is a sequence $x_1<\cdots <x_k$ such that, for $1<j<k$, $x_j \geq \frac{x_{j+1}+x_{j-1}}{2}$
Erdos781.f f k is the minimal n such that any $2$-colouring of $\{1,\ldots,n\}$ (identified with Fin n) contains a monochromatic $k$-term descending wave
Erdos781.erdos_781.variants.brown_erdos_freedman Brown, Erdős, and Freedman [BEF90] proved $k^2-k+1\leq f(k) \leq \frac{k^3-4k+9}{3}$
Erdos785.counting The counting function $A(x)=\lvert A\cap [1,x]\rvert$
Erdos785.aStar The largest element of $A$ in $[1,x]$, that is $a^*(x)=\max A\cap [1,x]$ (equal to $0$ when there is no such element)
Erdos785.IsExactAdditiveComplement Two sets $A, B\subseteq \mathbb{N}$ are exact additive complements if $A+B$ contains all large integers and $A(x)B(x)\sim x$
Erdos785.erdos_785.variants.danzer Danzer [Da64] proved that exact additive complements exist (Hanani had earlier conjectured they do not exist, as reported in [Er57] and [Er61])
Erdos785.erdos_785.variants.sarkozy_szemeredi Sárközy and Szemerédi [SaSz94] proved that it is impossible for $$A(x)B(x)-x=o(A(x)).$$
Erdos785.erdos_785.variants.chen_fang Chen and Fang [ChFa15] proved $A(x)B(x)-x\ll \min(A(x),B(x))^c$ cannot hold for any constant $c>0$
Erdos785.erdos_785.variants.narkiewicz Narkiewicz [Na59] proved that, under the given assumptions (and perhaps swapping $A$ and $B$) we must have $A(2x)/A(x)\to 1$ and $B(2x)/B(x)\to 2$
Erdos785.erdos_785.variants.ruzsa_upper_bound Ruzsa [Ru17] has constructed, for any function $w(x)\to \infty$, such a pair of sets with $$A(x)B(x)-x<w(x)$$ for infinitely many $x$
Erdos785.erdos_785.variants.ruzsa_lower_bound Documented declaration: ruzsa_lower_bound
Erdos785.erdos_785.variants.chen_fang_limsup Documented declaration: chen_fang_limsup
Erdos785.erdos_785.variants.chen_conjecture Chen conjectures that this should be true with $3/2$
Erdos785.erdos_785.variants.chen_fang_sharp This is sharp, as Chen and Fang [ChFa11] also proved that there exist such
Erdos786.Set.IsMulCardSet Nat.IsMulCardSet A means that A is a set of natural numbers that satisfies the property that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$
Erdos786.erdos_786.parts.i Let $\epsilon > 0$
Erdos786.erdos_786.parts.ii Is there some set $A\subset\{1, ..., N\}$ of size $\geq (1 - o(1))N$ such that $a_1\cdots a_r = b_1\cdots b_s$ with $a_i, b_j\in A$ can only hold when $r = s$?
Erdos786.erdos_786.parts.i.example An example of such a set with density $\frac 1 4$ is given by the integers $\equiv 2\pmod{4}$
Erdos786.consecutivePrimesFrom consecutivePrimesFrom p k gives the set of k + 1 consecutive primes that are at least p in size
Erdos786.erdos_786.parts.i.selfridge Let $\epsilon > 0$ be given
Erdos789.IsSubsetSumSeparatingCard Given a non-negative integer
Erdos789.subsetSumThreshold The subset sum threshold $h(n)$, for each positive $n$, is the maximal separating cardinality of subset sums for $n$
Erdos789.erdos_789 Documented declaration: erdos_789
Erdos789.erdos_789.variants.sq Documented declaration: sq
Erdos789.erdos_789.variants.isBigO_sq Straus [Str66] proved that $h(n) \ll \sqrt{n}$
Erdos789.erdos_789.variants.sq_isBigO By the solved variant erdos_789.variants.isBigO_sq, in order to prove erdos_789.variants.sq it suffices to show $\sqrt{n}=O(h(n))$
Erdos79.erdos_79 We say $G$ is Ramsey size linear if $R(G,H)\ll m$ for all graphs $H$ with $m$ edges and no isolated vertices
Erdos793.Strongly2Primitive A finite set A ⊆ ℕ is strongly 2-primitive if a ∤ b * c whenever a, b, c ∈ A with a ≠ b and a ≠ c
Erdos793.F F n is the maximal size of a strongly 2-primitive subset of {1, …, n}
Erdos793.erdos_793 Let $F(n)$ be the maximum possible size of a subset $A \subseteq \{1, \ldots, n\}$ such that $a \nmid bc$ whenever $a, b, c \in A$ with $a \neq b$ and $a \neq c$
Erdos794.harrisHypergraph Documented declaration: harrisHypergraph
Erdos794.erdos_794 Documented declaration: erdos_794
Erdos794.erdos_794.variants.balogh Documented declaration: balogh
Erdos794.erdos_794.variants.harris Documented declaration: harris
Erdos794.erdos_794.variants.frankl_furedi This problem is then now asking how many edges a $3$-uniform hypergraph can have before it contains $K_4$ minus an edge, and whether the critical edge density is $2/9$
Erdos796.repCount The number of unordered representations m = a₁ * a₂ by two distinct elements a₁ < a₂ of A
Erdos796.HasRepBound A has at most k - 1 representations of every m (fewer than k)
Erdos796.g g k n = g_k(n) is the largest size of a subset A ⊆ {1, …, n} in which every m has fewer than k representations m = a₁ a₂ with a₁ < a₂ ∈ A
Erdos796.normalizedError The proposed second-order rescaling of g_3(n): (g_3(n) - (log log n / log n) · n) / (n / log n), whose limit is the constant c in the problem
Erdos796.erdos_796 Let $k\geq 2$ and let $g_k(n)$ be the largest possible size of $A\subseteq \{1,\ldots,n\}$ such that every $m$ has $<k$ solutions to $m=a_1a_2$ with $a_1<a_2\in A$
Erdos798.gridPoints The grid $\{1,\ldots,n\}^2$, viewed as a set of points of the plane
Erdos798.IsLineCover S is a set of points of $\{1,\ldots,n\}^2$ such that the lines determined by the pairs of distinct points of S cover all points of $\{1,\ldots,n\}^2$
Erdos798.t The minimum number of points in $\{1,\ldots,n\}^2$ such that the lines determined by these points cover all points in $\{1,\ldots,n\}^2$
Erdos798.erdos_798 Let $t(n)$ be the minimum number of points in $\{1,\ldots,n\}^2$ such that the $\binom{t}{2}$ lines determined by these points cover all points in $\{1,\ldots,n\}^2$
Erdos798.erdos_798.variants.lower_bound A problem of Erdős and Purdy, who proved $t(n) \gg n^{2/3}$
Erdos798.erdos_798.variants.upper_bound Resolved by Alon [Al91] who proved $t(n) \ll n^{2/3}\log n$
Erdos80.bookNumber The size of the largest book in G: the greatest number of triangles sharing a single edge
Erdos80.EveryEdgeInTriangle Every edge of G lies in at least one triangle
Erdos80.Admissible The graphs the problem quantifies over: n vertices, at least $cn^2$ edges, and every edge in a triangle
Erdos80.f $f_c(n)$, the largest book size forced on every admissible graph, which is the least book number among them
Erdos80.erdos_80.parts.i Documented declaration: i
Erdos80.erdos_80.parts.ii The weaker question from the same problem: is $f_c(n) \gg \log n$?
Erdos80.bookNumber_eq_zero_iff A book is counted by triangles, so bookNumber is 0 exactly when no edge lies in one
Erdos80.bookNumber_bot On a graph with no edges there is no book
Erdos80.everyEdgeInTriangle_bot EveryEdgeInTriangle holds vacuously on a graph with no edges, so the admissible set is nonempty only through the edge count
Erdos800.erdos_800 If $G$ is a graph on $n$ vertices which has no two adjacent vertices of degree $\geq 3$ then $$R(G)\ll n,$$ where the implied constant is absolute
Erdos803.erdos_803.variants.alon Documented declaration: alon
Erdos803.erdos_803.variants.janzer_sudakov Documented declaration: janzer_sudakov
Erdos804.HasLocalIndependence A graph G on n vertices in which every induced subgraph on m vertices has an independent set of size at least t
Erdos804.f f m n is maximal such that any graph on n vertices in which every induced subgraph on m vertices has an independent set of size at least
Erdos806.erdos_806.variants.alon_bukh_sudakov Documented declaration: alon_bukh_sudakov
Erdos806.erdos_806.variants.erdos_newman Erdős and Newman [ErNe77] proved that there exist
Erdos81.erdos_81 Let $G$ be a chordal graph on $n$ vertices - that is, $G$ has no induced cycles of length greater than $3$
Erdos81.erdos_81.variants.split Chen, Erdős, and Ordman [CEO94] have shown that any split graph can be partitioned into $\frac{3}{16}n^2+O(n)$ many cliques
Erdos812.erdos_812.parts.i Is it true that $\frac{R(n+1)}{R(n)}\geq 1+c$ for some constant $c>0$, for all large $n$?
Erdos812.erdos_812.parts.ii Is it true that $R(n+1)-R(n) \gg n^2$?
Erdos812.erdos_812.variants.lower_bound Burr, Erdős, Faudree, and Schelp [BEFS89] proved that $R(n+1)-R(n) \geq 4n-8$ for all $n\geq 2$
Erdos813.EverySevenHasTriangle Every set of $7$ vertices of G contains a triangle, that is, three pairwise adjacent vertices
Erdos813.h h n is $h(n)$: the smallest clique number of a graph on $n$ vertices in which every set of $7$ vertices contains a triangle
Erdos813.everySevenHasTriangle_top The complete graph on $n$ vertices satisfies the condition
Erdos813.h_seven For $n = 7$ the graph must contain a triangle, and a triangle plus $4$ isolated vertices has clique number $3$, so $h(7) = 3$
Erdos813.erdos_813 Let $h(n)$ be minimal such that every graph on $n$ vertices where every set of $7$ vertices contains a triangle must contain a clique on at least $h(n)$ vertices
Erdos813.erdos_813.variants.erdos_hajnal Erdős and Hajnal proved (see [Er91]) that $$n^{1/3} \ll h(n) \ll n^{1/2}.$$
Erdos813.erdos_813.variants.bucic_sudakov Bucić and Sudakov [BuSu23] proved that $h(n) \gg n^{5/12-o(1)}$
Erdos813.erdos_813.variants.lower The lower half of Erdos813.erdos_813 has a positive answer: there is a constant $c_1 > 0$ with $n^{1/3+c_1} \ll h(n)$
Erdos813.erdos_813.variants.upper The upper half of Erdos813.erdos_813: is there a constant $c_2 > 0$ with $h(n) \ll n^{1/2-c_2}$?
Erdos814.erdos_814.variants.sauermann Sauermann [Sa19] proved this with $c_k \gg 1/k^3$
Erdos816.HasEqualDegreePathThree G contains two vertices of the same degree which are joined by a path of length 3
Erdos816.erdos_816.variants.chen_ma Chen and Ma [ChMa25] proved that, provided
Erdos816.erdos_816.variants.complete_bipartite The example of $K_{n,n+1}$ shows that this fails if we only have $n^2+n$ edges
Erdos817.erdos_817 Let $k \geq 3$
Erdos817.erdos_817.variants.bdd_power A problem of Erdős and Sárközy who proved $$ g_3(n) \gg \frac{3^n}{n^{O(1)}}. $$
Erdos818.erdos_818 Let $A$ be a finite set of integers such that $\lvert A+A\rvert \ll \lvert A\rvert$
Erdos818.erdos_818.variants.solymosi This was proved by Solymosi [So09d], in the strong form $$\lvert AA\rvert \gg \frac{\lvert A\rvert^2}{\log \lvert A\rvert}.$$
Erdos82.IsRegularInduced A predicate that holds if $S$ is a regular induced subgraph of $G$
Erdos82.F $F(n)$ is the maximal integer such that every graph on $n$ vertices contains a regular induced subgraph on at least $F(n)$ vertices
Erdos82.erdos_82 $F(n) / \log n \to \infty as n \to \infty$
Erdos82.erdos_82.variants.F_upper_bound $F(n) \le O(n^{1/2} \ln ^ {3/4} n)$ Theorem 1.4 from [AKS07] [AKS07] Alon, N
Erdos821.g Let $g(n)$ count the number of $m$ such that $\phi(m)=n$
Erdos821.erdos_821 Is it true that, for every $\epsilon>0$, there exist infinitely many $n$ such that $g(n) > n^{1-\epsilon}$?
Erdos821.erdos_821.variants.pillai Pillai proved that $\limsup g(n)=\infty$
Erdos821.erdos_821.variants.erdos Erdős [Er35b] proved that there exists some constant $c>0$ such that $g(n) > n^c$ for infinitely many $n$
Erdos821.erdos_821.variants.lichtman The best known bound is that there are infinitely many
Erdos822.erdos_822 Does the set of integers of the form $n + \varphi(n)$ have positive (lower) density?
Erdos825.erdos_825.variants.necessary_cond Show that if the constant $C > 0$ is such that every integer $n$ with $\sigma(n) > Cn$ is the distinct sum of proper divisors of $n$, then we must have $C > 2$
Erdos826.erdos_826 Are there infinitely many $n$ such that, for all $k\geq 1$ $$ \tau(n + k) \ll k? $$
Erdos828.erdos_828 Is it true that, for any $a \in \mathbb{Z}$, there are infinitely many $n$ such that $$\phi(n) | n + a$$?
Erdos828.erdos_828.variants.lehmer_conjecture When $n > 1$, Lehmer conjectured that $\phi(n) | n - 1$ if and only if $n$ is prime
Erdos829.cubes The set of perfect cubes in $\mathbb{N}$
Erdos829.mem_cubes_iff Membership in cubes can be witnessed by a bounded cube root, which makes it decidable for concrete values
Erdos829.erdos_829 Erdős Problem 829 (open)
Erdos829.sumRep_cubes_zero There is exactly one ordered pair of cubes summing to $0$, namely $(0, 0)$
Erdos829.sumRep_cubes_two The only ordered pair of cubes summing to $2$ is $(1, 1)$
Erdos829.sumRep_cubes_three The integer $3$ is not the sum of two cubes
Erdos829.sumRep_cubes_taxicab The Hardy-Ramanujan taxicab number satisfies $1729 = 1^3 + 12^3 = 9^3 + 10^3$, giving the four ordered representations $(1, 1728), (1728, 1), (729, 1000), (1000, 729)$
Erdos829.variants.mordell Mordell proved $\limsup_{n \to \infty} (1_A \ast 1_A)(n) = \infty$, where $A$ is the set of perfect cubes
Erdos829.variants.mahler Mahler proved $(1_A \ast 1_A)(n) \gg (\log n)^{1/4}$ for infinitely many $n$, where $A$ is the set of perfect cubes
Erdos829.variants.stewart Stewart improved Mahler's lower bound to $(1_A \ast 1_A)(n) \gg (\log n)^{11/13}$ for infinitely many $n$, where $A$ is the set of perfect cubes
Erdos83.erdos_83.variants.extremal The bound in erdős_83 is attained by the family of all subsets of $[4n]$ of size $2n$ containing at least $n+1$ elements from $[2n]$
Erdos83.erdos_83.variants.ahlswede_khachatrian Documented declaration: ahlswede_khachatrian
Erdos830.A Let $A(x)$ count the number of amicable pairs $1\leq a\leq b\leq x$
Erdos830.erdos_830.parts.i Erdos Problem 830, Part 1 We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$
Erdos830.erdos_830.parts.ii Erdos Problem 830, Part 2 We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$
Erdos830.erdos_830.variants.erdos We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$
Erdos830.erdos_830.variants.pomerance We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$
Erdos830.erdos_830.variants.pomerance_stronger We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$
Erdos832.erdos_832 Let $r\geq 3$ and $k$ be sufficiently large in terms of $r$
Erdos833.erdos_833 Does there exist an absolute constant
Erdos834.erdos_834.parts.i Documented declaration: i
Erdos834.erdos_834.parts.ii Documented declaration: ii
Erdos835.Property The property that for a given $k$, the $k$-subsets of a $2k$-set can be colored with $k+1$ colors such that any $(k+1)$-subset contains all colors
Erdos835.erdos_835 Documented declaration: erdos_835
Erdos835.erdos_835.variants.johnson Alternative statement of Erdős Problem 835 using the chromatic number of the Johnson graph
Erdos835.johnsonGraph_2k_k_chromaticNumber_known_cases It is known that for $3 \leq k \leq 8$, the chromatic number of $J(2k, k)$ is greater than $k+1$, see Johnson graphs
Erdos835.johnsonGraph_18_9_chromaticNumber The smallest case not on this page is $k=9$: But that one can be solved as well: The chromatic number of $J(18, 9)$ is at least $11$
Erdos835.johnsonBound Johnson's upper bound on the maximum size A(n, d, w) of a n-dimensional binary code of distance d and weight w is as follows: * If d > 2 * w, then A(n, d, w) = 1
Erdos835.indepNum_johnson_le_johnsonBound Johnson's bound for the independence number of the Johnson graph
Erdos835.div_johnsonBound_le_chromaticNum_johnson Johnson's bound for the chromatic number of the Johnson graph
Erdos835.chromaticNumber_johnson_2k_k_lower_bound It is known that for $3 \leq k \leq 8$, the chromatic number of $J(2k, k)$ is greater than $k+1$, see Johnson graphs
Erdos835.chromaticNumber_johnson_2k_k_lower_bound_odd It is also known that for $3 \leq k \leq 203$ odd, the chromatic number of $J(2k, k)$ is greater than $k+1$, see Johnson graphs
Erdos835.johnson_chromaticNumber_odd It can be seen that the chromatic number of $J(2k,k)$ is $>k+1$ for all odd $k>2$
Erdos835.johnson_chromaticNumber_composite Ma and Tang have proved that the chromatic number of $J(2k,k)$ is $>k+1$ for all $k>2$ not of the form $p-1$ for prime $p$
Erdos835.johnsonGraph_chromaticNumber_odd_of_johnson_chromaticNumber_composite Ma and Tang's result implies the cases for odd $k$
Erdos835.johnson_chromaticNumber Is the chromatic number of J(2 * k, k) always at least k + 2?
Erdos836.erdos_836.parts.i Let $r\geq 2$ and let $G$ be an intersecting $r$-uniform hypergraph with chromatic number $3$
Erdos836.erdos_836.parts.ii In an intersecting $r$-uniform hypergraph with chromatic number $3$, must there be two distinct edges which meet in $\gg r$ many vertices?
Erdos837.erdos_837 Documented declaration: erdos_837
Erdos839.SumOfConsecutiveFree A sequence $a : \mathbb{N} \to \mathbb{N}$ is "sum-of-consecutive-free" if no term equals the sum of a contiguous block of earlier terms
Erdos839.erdos_839.parts.i Documented declaration: i
Erdos839.erdos_839.parts.ii Documented declaration: ii
Erdos841.t t n is the least T such that {n + 1, …, n + T} contains a subset whose product with n is a square; t n = 0 if and only if n is itself a square
Erdos841.erdos_841 Let $t_n$ be minimal such that $\{n+1,\ldots,n+t_n\}$ contains a subset whose product with $n$ is a square number (and let $t_n=0$ if $n$ is itself square)
Erdos841.erdos_841.variants.selfridge_large_prime Selfridge proved that $t_n=P(n)$ if $P(n)>\sqrt{2n}+1$
Erdos841.erdos_841.variants.selfridge_sqrt Selfridge proved that $t_n \ll n^{1/2}$ if $P(n)\leq\sqrt{2n}+1$
Erdos841.erdos_841.variants.many_small_values Bui, Pratt, and Zaharescu [BPZ24] proved that for at least $x^{1-o(1)}$ many $n\leq x$ we have $$t_n \leq \exp(O(\sqrt{\log n\log\log n})).$$ This was formalized in Lean by Codex
Erdos841.erdos_841.variants.lower_bound Bui, Pratt, and Zaharescu [BPZ24] proved that for all non-square $n$ $$t_n \gg (\log\log n)^{6/5}(\log\log\log n)^{-1/5}.$$ This was formalized in Lean by Codex
Erdos842.IsTrianglesPlusHamiltonianCycle Documented declaration: IsTrianglesPlusHamiltonianCycle
Erdos844.evenOrOddNonSquarefree Those $n \in \{1,\ldots,N\}$ which are even or not squarefree, that is, the set of even numbers together with the odd non-squarefree numbers
Erdos845.erdos_845 Let $C > 0$
Erdos846.NonTrilinearFor We say a subset A of points in the plane is ε-non-trilinear if any subset B of A, contains a non-trilinear subset C of size at least ε|B|
Erdos846.WeaklyNonTrilinear We say a subset A of points in the plane is weakly non-trilinear if it is a finite union of non-trilinear sets
Erdos846.erdos_846 Documented declaration: erdos_846
Erdos847.HasFew3APs Documented declaration: HasFew3APs
Erdos848.NonSquarefreeProductProp A set $A$ has the non-squarefree product property if $ab + 1$ is not squarefree for all $a, b ∈ A$
Erdos848.A₇ The candidate extremal set: $\{n ∈ \{0, \dots, N-1\} : n ≡ 7 (mod 25)\}$
Erdos848.Erdos848For The Erdős Problem 848 statement for a fixed $N$: any set $A ⊆ \{0, \dots, N-1\}$ with the non-squarefree product property has cardinality at most $|A₇(N)|$
Erdos848.erdos_848 Is the maximum size of a set $A ⊆ \{1, \dots, N\}$ such that $ab + 1$ is never squarefree (for all $a, b ∈ A$) achieved by taking those $n ≡ 7 \pmod{25}$?
Erdos848.erdos_848.variants.asymptotic There exists $N₀$ such that for all $N ≥ N₀$, if $A ⊆ \{1, \dots, N\}$ satisfies that $ab + 1$ is never squarefree for all $a, b ∈ A$, then $|A| ≤ |\{n ≤ N : n ≡ 7 \pmod{25}\}|$
Erdos849.erdos_849 Is it true that, for every integer $t\geq1$, there is some integer $a$ such that ${n \choose k} = a$ with $1\leq k \le \frac{n}{2}$ has exactly $t$ solutions?
Erdos85.f Let $f(n)$ be the smallest integer for which every graph on $n$ vertices with minimal degree $\geq f(n)$ contains a $C_4$
Erdos85.erdos_85 Is it true that, for all large $n$, $f(n + 1) \ge f(n)$?
Erdos850.erdos_850 Can there exist two distinct integers
Erdos851.TwoPowAddSet TwoPowAddSet r is the set of integers of the form 2^k+n, where k ≥ 0 and n has at most r prime divisors
Erdos851.twoPowAddPrimeSet The set of integers of the form 2^k+p, where k ≥ 0 and p is prime
Erdos851.erdos_851.variants.romanoff The set of integers of the form 2^k+p (where p is prime) has positive lower density
Erdos853.r Let r(x) be the smallest even integer t such that primeGap = t has no solutions for n ≤ x
Erdos853.erdos_853.parts.i Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime
Erdos853.erdos_853.parts.ii Let $d_n = p_{n+1} - p_n$, where $p_n$ is the $n$th prime
Erdos855.erdos_855 Erdős Problem 855 (Segal's conjecture): $\pi(x + y) \le \pi(x) + \pi(y)$ for all sufficiently large $x, y$, i.e. for all $x, y \ge N$ for some $N$
Erdos857.m m(n, k): minimal sunflower-forcing family size in the non-uniform [n] model
Erdos857.erdos_857 Estimate m(n,k), or better give an asymptotic formula
Erdos858.IsAdmissible A has no solution to $at=b$ with $a,b\in A$ and the smallest prime factor of $t$ greater than $a$
Erdos858.M The maximum of $\sum_{n\in A}\frac{1}{n}$ over all admissible $A\subseteq \{1,\ldots,N\}$
Erdos858.erdos_858.variants.positive The limit $c\approx 0.618\cdots$ is positive: for any fixed large $N$ the maximum is bounded away from $0$ (Alexander [Al66], Erdős–Sárközi–Szemerédi [ESS68])
Erdos858.erdos_858.variants.infinite Alexander [Al66] and Erdős, Sárközi, and Szemerédi [ESS68] proved that if $A$ is an infinite set with this property then $\sum_{n\in A\cap [1,N]}\frac{1}{n}=o(\log N)$
Erdos859.DivisorSumSet DivisorSumSet t is the set of natural numbers n such that t can be represented as a sum of distinct divisors of n
Erdos859.erdos_859.variants.erdos_upper_lower_bounds Documented declaration: erdos_upper_lower_bounds
Erdos859.erdos_859 The density of the divisor sum set is asymptotically equivalent to $c_1 / \log(t)^{c_2}$
Erdos859.erdos_859.variants.trivial_case A case where we can easily calculate the density of DivisorSumSet t is that of t=0
Erdos859.erdos_859.variants.positive_density An easy sanity check is to prove that for every natural number t the density dₜ is a positive number
Erdos86.erdos_86 Let $Q_n$ be the $n$-dimensional hypercube graph (so that $Q_n$ has $2^n$ vertices and $n2^{n-1}$ edges)
Erdos862.numMaximalSidonSets $A_1(N)$, the number of maximal Sidon subsets of $\{1, \dots, N\}$
Erdos862.erdos_862.parts.i Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$
Erdos862.erdos_862.parts.ii Let $A_1(N)$ be the number of maximal Sidon subsets of $\{1,\ldots,N\}$
Erdos862.erdos_862.variants.lower_bound This is resolved as a consequence of results of Saxton and Thomason [SaTh15] - they prove that the number of Sidon sets in $\{1,\ldots,N\}$ is at least $2^{(1.16+o(1))N^{1/2}}$
Erdos865.erdos_865.variants.k2 It is a classical folklore fact that if $A\subseteq \{1,\ldots,2N\}$ has size $\geq N+2$ then there are distinct $a,b\in A$ such that $a+b\in A$, which establishes the $k=2$ case
Erdos865.erdos_865.variants.sos Documented declaration: sos
Erdos865.erdos_865.variants.upper_bound Choi, Erdős, and Szemerédi [CES75] have proved that, for all
Erdos867.ConsecutiveSumFree A finite set of naturals
Erdos867.erdos_867 Documented declaration: erdos_867
Erdos867.erdos_867.variants.lower_bound Taking $A=(N/2,N]\cap \mathbb{N}$ shows $\lvert A\rvert \geq N/2-O(1)$ is possible
Erdos867.erdos_867.variants.adenwalla Adenwalla has observed that $$\lvert A\rvert \leq (\tfrac{2}{3}+o(1))N.$$
Erdos867.erdos_867.variants.freud Freud [Fr93] constructed a sequence with density $\geq 19/36$
Erdos867.erdos_867.variants.coppersmith_phillips_lower_bound The current best bounds are due to Coppersmith and Phillips [CoPh96], who prove that the maximal size of such an $A$ satisfies $$\frac{13}{24}N -O(1)\leq \lvert A\rvert.$$
Erdos867.erdos_867.variants.coppersmith_phillips_upper_bound Documented declaration: coppersmith_phillips_upper_bound
Erdos868.erdos_868.parts.i Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$
Erdos868.erdos_868.parts.ii Let $A$ be an additive basis of order $2$, let $f(n)$ denote the number of ways in which $n$ can be written as the sum of two elements from $A$
Erdos868.erdos_868.variants.fixed_ε Erdős and Nathanson proved that this is true if $f(n) > c \log n$ for all large $n$, for some fixed constant $c > (\log \frac{4}{3})^{-1}$
Erdos87.erdos_87.parts.i Let $0 < \epsilon < 1$
Erdos87.erdos_87.parts.ii Even stronger, is there some $c > 0$ such that, for all large $k$, $$R(G) > c R(k)$$ for every graph $G$ with chromatic number $\chi(G)=k$?
Erdos871.erdos_871 Let $A$ be an additive basis of order $2$, and suppose $1_A\ast 1_A(n)\to \infty$ as $n\to \infty$
Erdos872.IsPrimitive A primitive subset of $\{2, \dots, n\}$ is a set in which no element divides another
Erdos872.GamePos A game position consists of the already claimed set and the still unclaimed pool
Erdos872.legalMoves The legal moves from a position: unclaimed elements whose insertion preserves primitiveness
Erdos872.mem_legalMoves Membership in legalMoves: a legal move is a pool element whose insertion preserves primitiveness
Erdos872.applyMove Apply a move by claiming x and removing it from the unclaimed pool
Erdos872.startPos The empty starting position on $\{2, \dots, n\}$
Erdos872.gameValueAux Auxiliary finite minimax recursion with an explicit fuel bound
Erdos872.gameValueAux_le Each move claims exactly one pool element, so the minimax value never exceeds the number of already claimed elements plus the number of still unclaimed elements
Erdos872.gameLength The minimax value of the primitive-set saturation game from p, with Prolonger to move
Erdos872.L The Erdős primitive-set game length on $\{2, \dots, n\}$
Erdos872.erdos_872.parts.i Erdős Problem 872, part (i) (weak form): there exists a constant $\epsilon > 0$ such that the game length is at least $\epsilon \cdot n$ for all sufficiently large $n$
Erdos872.erdos_872.parts.ii Erdős Problem 872, part (ii) (strong form): for every $\epsilon > 0$, the game length is at least $(1-\epsilon) \cdot n / 2$ for all sufficiently large $n$
Erdos872.erdos_872.trivial_upper_bound A trivial upper bound: a play can claim at most the $n - 1$ elements of $\{2, \dots, n\}$, so $L(n) \leq n - 1$
Erdos872.erdos_872.variants.prime_question Forum-related variant: how small can a maximal primitive subset of $\{2, \dots, n\}$ be?
Erdos873.F Let $a$ be some sequence of natural numbers
Erdos873.erdos_873 Documented declaration: erdos_873
Erdos873.erdos_873.variants.triple_upper_bound The upper bound (2)
Erdos873.erdos_873.variants.triple_lower_bound_infinitely_often The infinitely-often lower bound (3)
Erdos877.IsMaximalSumFree A is a maximal sum-free subset of {1, …, n}
Erdos877.fm fm n counts the number of maximal sum-free subsets $A\subseteq\{1,\ldots,n\}$
Erdos877.erdos_877.variants.cameron_erdos Cameron and Erdős [CaEr90] proved that $f_m(n)>2^{n/4}$
Erdos877.erdos_877.variants.balogh_liu_sharifzadeh_treglown Balogh, Liu, Sharifzadeh, and Treglown [BLST15] proved that $f_m(n)=2^{(\frac{1}{4}+o(1))n}$
Erdos881.IsMinimalAsymptoticAddBasisOfOrder We interpret "additive basis of order k" as an asymptotic additive basis of order k, using the predicate Set.IsAsymptoticAddBasisOfOrder from additive combinatorics
Erdos881.erdos_881 Let A ⊂ ℕ be an additive basis of order k which is minimal in the sense that if B ⊂ A is any infinite set, then A \ B is not a basis of order k
Erdos883.coprimeGraph The coprime graph on $\mathbb{N}$: two integers are joined by an edge if they are coprime
Erdos883.erdos_883.parts.i For $A\subseteq \{1,\ldots,n\}$ let $G(A)$ be the graph with vertex set $A$, where two integers are joined by an edge if they are coprime
Erdos883.erdos_883.parts.ii Documented declaration: ii
Erdos884.sumDivisorInvPairwiseDifference The sum $\sum_{1 \le i < j \le \tau(n)} \frac{1}{d_j - d_i}$ over all pairs of divisors $d_i < d_j$ of $n$
Erdos884.sumDivisorInvConsecutiveDifference The sum $\sum_{1 \le i < \tau(n)} \frac{1}{d_{i + 1} - d_i}$ over consecutive divisors of $n$
Erdos884.Erdos884Prop For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of $n$ in increasing order
Erdos884.erdos_884 For a natural number n, let $1 = d_1 < \dotsc < d_{\tau(n)} = n$ denote the divisors of $n$ in increasing order
Erdos884.erdos_884_false_of_hardy_littlewood Documented declaration: erdos_884_false_of_hardy_littlewood
Erdos885.factorDifferenceSet For integer $n \geq 1$ we define the factor difference set of $n$ by $D(n) = \{|a-b| : n=ab\}$
Erdos885.erdos_885 Is it true that, for every $k \geq 1$, there exist integers $N_1 < \dots < N_k$ such that $|\cap_i D(N_i)| \geq k$?
Erdos885.erdos_885.variants.k_eq_2 Erdős and Rosenfeld [ErRo97] proved this is true for $k=2$
Erdos885.erdos_885.variants.k_eq_3 Jiménez-Urroz [Ji99] proved this for $k=3$
Erdos885.erdos_885.variants.k_eq_4 Bremner [Br19] proved this for $k=4$
Erdos886.Erdos886Divisors The set of divisors of $n$ in the interval $(n^{1/2}, n^{1/2} + n^{1/2-\epsilon})$
Erdos886.erdos_886 Let $\epsilon>0$
Erdos886.erdos_886.variants.rosenfeld_infinite Erdős and Rosenfeld [ErRo97] proved that there are infinitely many $n$ such that there are four divisors of $n$ in $(n^{1/2},n^{1/2}+16n^{1/4})$
Erdos886.erdos_886.variants.rosenfeld_bound Erdős and Rosenfeld [ErRo97] proved that, for any constant $C>0$, all large $n$ have at most $1+C^2$ many divisors in $[n^{1/2}, n^{1/2}+Cn^{1/4}]$
Erdos887.erdos_887.parts.i Is there an absolute constant
Erdos887.erdos_887.parts.ii Is there an absolute constant
Erdos887.erdos_887.variants.rosenfeld_infinite A question of Erdős and Rosenfeld, who proved that there are infinitely many $n$ with (at least) $4$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + cn^{\frac{1}{4}})$
Erdos887.erdos_887.variants.rosenfeld_4 Erdős and Rosenfeld, ask whether $4$ is the best possible $K$ for the infinitude of $n$ with (at least) $K$ divisors in $(n^{\frac{1}{2}}, n^{\frac{1}{2}} + n^{\frac{1}{4}})$
Erdos888.RequiredCondition Condition on the sets $A$ appearing in Erdős 888
Erdos888.p Proposition that for a specific $n$ an $A$ with the above defined condition and cardinality $k$ exists
Erdos888.erdos_888.variants.sarkozy Erdős claims that Sárközy proved that $\lvert A\rvert =o(n)$ (a proof of this bound is provided by Tao in the comments)
Erdos888.erdos_888.variants.primes The primes show that $\lvert A\rvert \gg n/\log n$ is possible
Erdos888.erdos_888.variants.semiprimes Cambie and Weisenberg have noted in the comments that the set of semiprimes also works, showing $(1+o(1))\frac{\log\log n}{\log n}n \leq \lvert A\rvert$ is achievable
Erdos889.v $v(n,k)$ counts the prime factors of $n+k$ which do not divide $n+i$ for all $0 \le i < k$
Erdos889.v₀ $v_0(n)$ is the supremum of $v(n,k)$ for all $k \ge 0$
Erdos889.erdos_889 Let $v(n,k)$ count the prime factors of $n+k$ which do not divide $n+i$ for $0\leq i < k$
Erdos889.erdos_889.variants.v0_gt_1 $v_0(n) > 1$ for all $n$ except $n$ = 0, 1, 2, 3, 4, 7, 8, 16 [ErSe67] Erdős, P
Erdos889.v_l $v_l(n)$ is the supremum of $v(n,k)$ for all $k \ge l$
Erdos889.erdos_889.variants.general Let $v_l(n) = \max_{k\geq l} v(n,k)$
Erdos889.erdos_889.variants.v1_eq_1_finite Does $v_1(n) = 1$ have finite solutions?
Erdos889.V Documented declaration: V
Erdos889.V_l $V_l(n)$ is the supremum of $V(n,k)$ for all $k \ge l$
Erdos89.erdos_89 Erdős [Er46] asked whether every set of $n$ distinct points in $\mathbb{R}^2$ determines $\gg \frac{n}{\sqrt{\log n}}$ many distinct distances
Erdos89.erdos_89.variants.n_dvd_log_n Guth and Katz [GuKa15] proved that there are always $\gg \frac{n}{\log n}$ many distinct distances
Erdos89.erdos_89.variants.grid_upper_bound Documented declaration: grid_upper_bound
Erdos89.erdos_89.variants.implies_n_dvd_log_n This theorem provides a sanity check, showing that the main conjecture (erdos_89) is strictly stronger than the solved Guth and Katz result
Erdos890.omegaGt omegaGt k n counts the number of distinct prime factors of n that are strictly greater than k
Erdos890.erdos_890.parts.a Documented declaration: a
Erdos890.erdos_890.parts.b Documented declaration: b
Erdos890.erdos_890.variants.liminf_lower_bound A question of Erdős and Selfridge [ErSe67], who observe that $\liminf_{n\to \infty}\sum_{0\leq i < k}\omega(n+i)\geq k+\pi(k)-1$ for every $k$
Erdos890.erdos_890.variants.omega_limsup It is a classical fact that $\limsup_{n\to \infty}\omega(n)\frac{\log\log n}{\log n}=1.$
Erdos891.erdos_891 Let $2=p_1 < p_2 < \cdots$ be the primes and $k\geq 2$
Erdos891.erdos_891.variants.schinzel Documented declaration: schinzel
Erdos891.erdos_891.variants.case_k_2 This is unknown even for $k=2$ - that is, is it true that in every interval of $6$ (sufficiently large) consecutive integers there must exist one with at least $3$ prime factors?
Erdos891.erdos_891.variants.weisenberg Weisenberg has observed that Dickson's conjecture implies the answer is no if we replace $p_1\cdots p_k$ with $p_1\cdots p_k-1$
Erdos893.f Definition of function $f(n) := \sum_{1\leq k\leq n}\tau(2^k-1)$
Erdos893.erdos_893 Does the limit $\lim_{n\to\infty} \frac{f(2n)}{f(n)}$ tend to infinity?
Erdos893.erdos_893.variants.unbounded Documented declaration: unbounded
Erdos895.erdos_895.variants.hindman Hajnal thought that there is in fact an independent set which is a Hindman set - that is, an independent set of the shape
Erdos897.erdos_897.variants.parts.i Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$
Erdos897.erdos_897.variants.parts.ii Let $f(n)$ be an additive function (so that $f(ab)=f(a)+f(b)$ if $(a,b)=1$) such that $\limsup_{p,k} f(p^k) / \log(p^k) = ∞$ and $f(p^k) = f(p)$ or $f(p^k) = kf(p)$
Erdos898.erdos_898 Documented declaration: erdos_898
Erdos9.Erdos9A The set of odd numbers that cannot be expressed as a prime plus two powers of 2
Erdos9.erdos_9.variants.infinite The set is known to be infinite
Erdos9.erdos_9 Is the upper density of the set of odd numbers that cannot be expressed as a prime plus two powers of 2 positive?
Erdos90.unitDistanceCounts The set of all possible numbers of unit distances for a configuration of $n$ points
Erdos90.unitDistanceCounts_BddAbove This lemma confirms that the set of possible unit distance counts is bounded above, which ensures that taking the supremum (sSup) is a well-defined operation
Erdos90.maxUnitDistances The maximum number of unit distances determined by any set of $n$ points in the plane
Erdos90.erdos_90 Does every set of $n$ distinct points in $\mathbb{R}^2$ contain at most $n^{1+O(\frac{1}{\log\log n})}$ many pairs which are distance $1$ apart?
Erdos90.erdos_90.variants.sawin_explicit Sawin's explicit exponent
Erdos90.erdos_90.variants.sawin_explicit_implies_polynomial_lower_bound Sawin's explicit bound implies the qualitative polynomial lower bound, by taking $c = 0.014114$
Erdos90.erdos_90.variants.polynomial_lower_bound_implies_erdos_90 The polynomial lower bound implies the answer to Erdős 90 is False: a fixed positive exponent $c$ is incompatible with the conjectured $O(1 / \log \log n)$ growth
Erdos90.sawin_lattice_reduction Sawin's Lemma 2: lattice geometry of unit distances (Sawin, arXiv:2605.20579)
Erdos901.erdos_901 Let $m(n)$ be minimal such that there is an $n$-uniform hypergraph with $m(n)$ edges which is $3$-chromatic
Erdos903.erdos_903 Documented declaration: erdos_903
Erdos904.n An abbreviation for the fixed number of vertices $n$ in the graph
Erdos904.turanNumber The number of edges of the Turán graph $T(n, r)$, i.e. the Turán number
Erdos904.erdos_904 Let $r\geq 2$ and let $t_r(n)$ be the Turán number (the maximal number of edges in a graph on $n$ vertices with no $K_{r+1}$)
Erdos905.erdos_905 Every graph with $n$ vertices and $>n^2/4$ edges contains an edge which is in at least $n/6$ triangles
Erdos906.erdos_906 Does there exists an entire non-zero transcendental function f : ℂ → ℂ such that for any sequence n₀ < n₁ < ..., { z | ∃ k, iteratedDeriv (n k) f z = 0 } is dense
Erdos907.erdos_907 Let $f:\mathbb{R}\to \mathbb{R}$ be such that $f(x+h)-f(x)$ is continuous for every $h>0$
Erdos91.IsOptimal A set $A$ is 'optimal' if it has $n$ points and achieves the minimum distance count
Erdos91.DilationEquivSimilar Two finite sets of points in $\mathbb{R}^2$ are similar if one can be mapped to the other by a DilationEquiv
Erdos91.equiTriangle Equilateral triangle with unit side length, resting on the x-axis with one vertex at the origin
Erdos91.circleSeven Regular 7-gon with unit side length, touching both axes in the first quadrant
Erdos91.wheelSeven Wheel graph on 7 vertices (center + regular hexagon) with unit side length, touching both axes in the first quadrant
Erdos91.UniqueMinimizer Documented declaration: UniqueMinimizer
Erdos91.erdos_91 Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$
Erdos91.erdos_91.variants.three For $n = 3$ the equilateral triangle is the only such set
Erdos91.erdos_91.variants.four For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples
Erdos91.erdos_91.variants.five For $n = 5$ the regular pentagon is the unique such set (which has two distinct distances)
Erdos91.erdos_91.variants.six In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 6$
Erdos91.erdos_91.variants.seven In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 7$
Erdos91.erdos_91.variants.eight In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 8$
Erdos91.erdos_91.variants.nine In [Er87b] on p.171 Erdős says that there are at least two non-similar examples for $n = 9$
Erdos912.h If $n! = \prod_{i}p_i^{k_i}$ is the factorization into distinct primes, then we define $h(n)$ to be the number of distinct exponents $k_i$
Erdos912.erdos_912.variants.selfridge Erdős and Selfridge prove in [Er82c] that $h(n) \asymp \left(\frac{n}{\log n}\right)^{1/2}$
Erdos912.erdos_912 Prove that there exists some $c>0$ such that $$h(n) \sim c \left(\frac{n}{\log n}\right)^{1/2}$$ as $n\to \infty$
Erdos912.erdos_912.variants.tao A heuristic of Tao using the Cramér model for the primes suggests this is true with $c=\sqrt{2\pi}$
Erdos913.erdos_913 Are there infinitely many $n$ such that if $$ n(n + 1) = \prod_i p_i^{k_i} $$ is the factorisation into distinct primes then all exponents $k_i$ are distinct?
Erdos913.erdos_913.variants.infinite_many_8p_sq_add_one_primes It is likely that there are infinitely many primes $p$ such that $8p^2 - 1$ is also prime
Erdos913.erdos_913.variants.conditional If there are infinitely many primes $p$ such that $8p^2 - 1$ is prime, then this is true
Erdos914.erdos_914 Let $r\geq 2$ and $m\geq 1$
Erdos914.erdos_914.variants.equitable_colouring Documented declaration: equitable_colouring
Erdos915.erdos_915.variants.edge_disjoint The edge-disjoint reading of erdős_915
Erdos918.erdos_918.parts.i Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$?
Erdos918.erdos_918.parts.ii Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?
Erdos918.erdos_918.variants.all_subgraphs.parts.i Is there a graph with $\aleph_2$ vertices and chromatic number $\aleph_2$ such that every subgraph on $\aleph_1$ vertices has chromatic number $\leq\aleph_0$?
Erdos918.erdos_918.variants.all_subgraphs.parts.ii Is there a graph with $\aleph_{\omega+1}$ vertices and chromatic number $\aleph_1$ such that every subgraph on $\aleph_\omega$ vertices has chromatic number $\leq\aleph_0$?
Erdos918.erdos_918.variants.erdos_hajnal A question of Erd\H{o}s and Hajnal [ErHa68b], who proved, assuming the generalized continuum hypothesis, that for every finite
Erdos918.erdos_918.variants.eq_aleph_0.parts.i In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$
Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.i In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$
Erdos918.erdos_918.variants.eq_aleph_0.parts.ii In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$
Erdos918.erdos_918.variants.eq_aleph_0_all_subgraphs.parts.ii In [Er69b] the questions are stated with $= \aleph_0$ rather than $\leq\aleph_0$
Erdos92.maxEquidistantPointsAt For a given point x and a set of other points, this function finds the maximum number of points that lie on a single circle centered at x
Erdos92.hasMinEquidistantProperty This property holds for a set of points A if every point x in A has at least k other points from A that are equidistant from x
Erdos92.possible_f_values The set of all possible values k for which there exists a set of n points satisfying the hasMinEquidistantProperty k
Erdos92.possible_f_values_BddAbove A sanity check to ensure the set of possible f(n) values is bounded above
Erdos92.f Let $f(n)$ be maximal such that there exists a set $A$ of $n$ points in $\mathbb^2$ in which every $x \in A$ has at least $f(n)$ points in $A$ equidistant from $x$
Erdos92.erdos_92.variants.weak Is it true that $f(n)\leq n^{o(1)}$?
Erdos92.erdos_92.variants.strong Or even $f(n) < n^{c/\log\log n}$ for some constant $c > 0$?
Erdos920.f $f_k(n)$ is the maximum possible chromatic number of a graph with $n$ vertices which contains no $K_k$
Erdos920.erdos_920.variants.upper_bound Graver and Yackel [GrYa68] proved that $f_k(n) \ll \left(n\frac{\log\log n}{\log n}\right)^{1-\frac{1}{k-1}}.$
Erdos920.erdos_920.variants.k_eq_3 It is known that $f_3(n)\asymp (n/\log n)^{1/2}$ (see [erdosproblems.com/1104])
Erdos920.erdos_920.variants.lower_bound_k_eq_4 The lower bound $R(4,m) \gg m^3/(\log m)^4$ of Mattheus and Verstraete [MaVe23] (see [erdosproblems.com/166]) implies $f_4(n) \gg \frac{n^{2/3}}{(\log n)^{4/3}}$
Erdos920.erdos_920.variants.lower_bound_k_ge_5 A positive answer to this question for all
Erdos923.erdos_923 Is it true that, for every $k$, there is some $f(k)$ such that if $G$ has chromatic number $\geq f(k)$ then $G$ contains a triangle-free subgraph with chromatic number $\geq k$?
Erdos926.H The graph $H_k$ on the vertices $x, y_1, \ldots, y_k, z_{ij}$ ($i < j$): $x$ is adjacent to every $y_i$, and each pair $y_i, y_j$ is adjacent to the vertex $z_{ij}$
Erdos926.erdos_926.variants.aks Alon, Krivelevich, and Sudakov [AKS03] proved $\mathrm{ex}(n;H_k) \ll kn^{3/2}$ with an absolute implied constant
Erdos927.g g n is the maximum number of different sizes of cliques (maximal complete subgraphs) of a graph on n vertices
Erdos93.erdos_93 If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then they determine at least $\lfloor \frac{n}{2}\rfloor$ distinct distances
Erdos930.IsPower $n$ is a perfect power if there exist natural numbers $m$ and $l$ such that $1 < l$ and $m^l = n$
Erdos930.erdos_930 Documented declaration: erdos_930
Erdos930.nextPrime Returns the least prime satisfying $k \le p$
Erdos930.erdos_930.variants.consecutive_strong Let $k$, $l$, $n$ be integers such that $k \ge 3$, $l \ge 2$ and $n + k \ge p^{(k)}$, where $p^{(k)}$ is the least prime satisfying $p^{(k)} \ge k$
Erdos930.erdos_930.variants.consecutive_integers Erdos and Selfridge [ErSe75] proved that the product of consecutive integers is never a power (establishing the case $r=1$)
Erdos931.erdos_931 Let $k_1 \geq k_2 \geq 3$
Erdos931.erdos_931.variants.additional_condition Erdős thought perhaps if the two products have the same factors then $n_2 > 2(n_1 + k_1)$
Erdos931.erdos_931.variants.additional_condition_nonempty In fact there exist counterexamples, like this one found by AlphaProof
Erdos931.erdos_931.variants.exists_prime Erdős was unable to prove that if the two products have the same factors then there must exist a prime between $n_1$ and $n_2$
Erdos932.erdos_932 Let $p_k$ denote the $k$th prime
Erdos932.erdos_932.variants.one_le Erdős could show that the density of $r$ such that at least one such $n$ exists is $0$
Erdos933.k The 2-adic valuation of $n(n+1)$
Erdos933.l The 3-adic valuation of $n(n+1)$
Erdos933.erdos_933 If $n(n+1)=2^k3^lm$, where $(m,6)=1$, then is it true that $\limsup_{n\to \infty} \frac{2^k3^l}{n\log n}=\infty$?
Erdos933.erdos_933.variants.mahler Mahler proved (a more general result that implies in particular) that $2^k3^l<n^{1+o(1)}$
Erdos933.erdos_933.variants.lower_bound Erdős [Er76d] wrote 'it is easy to see' that for infinitely many $n$, $2^k 3^l > n\log n$
Erdos936.EventuallyNotPowerful The predicate that a n is only powerful for finitely many n
Erdos936.erdos_936.variants.two_pow_add_one Is $2^n + 1$ powerful for finitely many $n$?
Erdos936.erdos_936.variants.two_pow_sub_one Is $2^n - 1$ powerful for finitely many $n$?
Erdos936.erdos_936.variants.factorial_add_one Is $n! + 1$ powerful for finitely many $n$?
Erdos936.erdos_936.variants.factorial_sub_one Is $n! - 1$ powerful for finitely many $n$?
Erdos937.IsCoprimePowerfulAP4 The four numbers $a, a+d, a+2d, a+3d$ form a four-term arithmetic progression ($d > 0$) of pairwise coprime powerful numbers
Erdos937.not_isCoprimePowerfulAP4_zero_one Sanity check for IsCoprimePowerfulAP4: the progression $0, 1, 2, 3$ is not a valid example, since $2$ is not powerful
Erdos938.erdos_938 Let $A=\{n_1 < n_2 < \cdots\}$ be the sequence of powerful numbers (if $p\mid n$ then $p^2\mid n$)
Erdos939.Erdos939Sums A set S belongs to Erdos939Sums r if it meets the following criteria: - The elements are positive
Erdos939.erdos_939 If $r≥4$ then can the sum of $r-2$ coprime $r$-powerful numbers ever be itself $r$-powerful?
Erdos939.erdos_939.variants.finite If $r≥4$, are there at most finitely many sums of $r-2$ coprime $r$-powerful numbers that are themselves $r$-powerful?
Erdos939.erdos_939.variants.infinite_of_six_le For every $r \ge 6$ there are infinitely many sums of $r - 2$ coprime $r$-powerful numbers that are themselves $r$-powerful
Erdos939.erdos_939.variants.triples Are there infinitely many triples of coprime $3$-powerful numbers $a, b, c$ such that $a + b = c$?
Erdos939.erdos_939.variants.examples Cambie has found several examples of the sum of $r - 2$ coprime $r$-powerful numbers being itself $r$-powerful
Erdos939.erdos_939.variants.seven Cambie has also found solutions when $r=7$
Erdos939.erdos_939.variants.eight Cambie has also found solutions when $r=8$
Erdos939.erdos_939.variants.euler Documented declaration: euler
Erdos94.regularNGon The regular $n$-gon inscribed in the unit circle
Erdos94.erdos_94 Suppose $n$ points in $\mathbb{R}^2$ determine a convex polygon and the set of distances between them is $\{u_1,\ldots,u_t\}$
Erdos94.erdos_94.variants.sum_multiplicity Note it is trivial that $\sum f(u_i)=\binom{n}{2}$
Erdos94.erdos_94.variants.no_three_on_a_line Lefmann and Theile [LeTh95] prove a stronger version of this question, that $$\sum_i f(u_i)^2 \ll n^3$$ under the weaker assumption that no three points are on a line
Erdos94.erdos_94.variants.regular_ngon Erdős and Fishburn also make the stronger conjecture that $\sum f(u_i)^2$ is maximal for the regular $n$-gon (for large enough $n$)
Erdos940.erdos_940 Let $r \ge 3$
Erdos940.erdos_940.variants.two The set of integers which are the sum of at most two $2$-powerful numbers has density $0$
Erdos940.erdos_940.variants.three_cubes Is it true that the set of integers which are the sum of at most three cubes has density $0$?
Erdos940.erdos_940.variants.large_integers Let $r \ge 3$
Erdos940.erdos_940.variants.three_powerful Heath-Brown [He88] has proved that all large numbers are the sum of at most three $2$-powerful numbers
Erdos942.erdos_942.h Let $h(n)$ count the number of powerful integers in $[n^2, (n + 1)^2)$
Erdos942.erdos_942 Is there some constant $c > 0$ such that $h(n) < (\log n)^{c + o(1)}$ and, for infinitely many $n$, $h(n) > (\log n)^{c - o(1)}$
Erdos942.erdos_942.variants.limsup It is not hard to prove that $\limsup h(n) = \infty$
Erdos942.erdos_942.variants.density It is not hard to prove that the density $\delta_l$ of integers for which $h(n) = l$ exists and satisfies $$\sum_l \delta_l = 1$$
Erdos943.erdos_943 Let $A$ be the set of powerful numbers
Erdos944.SimpleGraph.IsErdos944 The predicate that graph $G$ with chromatic number $k$ is such that every vertex is critical, yet every critical set of edges has size $>r$
Erdos944.erdos_944 Let $k \ge 4$ and $r\ge 1$
Erdos944.erdos_944.variants.dirac_conjecture Let $k \ge 4$
Erdos944.erdos_944.variants.dirac_conjecture.k_eq_5 Documented declaration: k_eq_5
Erdos944.erdos_944.variants.dirac_conjecture.k_sub_one_not_prime Lattanzio [La02] proved there exist $k$-critical graphs without critical edges for all $k$ such that $k - 1$ is not prime
Erdos944.erdos_944.variants.dirac_conjecture.k_ge_five Jensen [Je02] gave an construction for $k$-critical graphs without any critical edges for all $k ≥ 5$
Erdos944.erdos_944.variants.large_k_for_any_r Documented declaration: large_k_for_any_r
Erdos945.F Let $F(x)$ be the maximal $k$ such that there exist $n+1, \dots, n+k \le x$ with $τ(n+1), \dots, τ(n+k)$ all distinct, where $τ(m)$ counts the divisors of $m$
Erdos945.Erdos945Prop Implementation note: we define a Prop here and below to be able to easily formulate the equivalence between the two variants
Erdos945.erdos_945 Is it true that $F(x) \leq (\log x)^{O(1)}$?
Erdos945.erdos_945.variants.constant Is there a constant $C > 0$ such that, for all large $x$, every interval $[x, x+(\log x)C]$ contains two integers with the same number of divisors?
Erdos945.erdos_945.variants.equivalence The two ways of phrasing the conjecture are equivalent
Erdos945.erdos_945.variants.lower_bound Erdős and Mirsky [ErMi52] proved that $\frac{(\log x)^{1/2}}{\log\log x}\ll F(x)$
Erdos945.erdos_945.variants.upper_bound Erdős and Mirsky [ErMi52] proved that $\log F(x) \ll \frac{(\log x)^{1/2}}$
Erdos946.erdos_946.variants.spiro_5040 There are infinitely many $n$ such that $τ(n) = τ(n + 5040)$
Erdos946.erdos946Count Number of $n \le x$ with $τ(n) = τ(n+1)$
Erdos946.erdos_946.variants.heathbrown_lower_bound The number of $n \le x$ with $τ(n) = τ(n+1)$ is at least $x / (\log x)^7$ for all sufficiently large $x$
Erdos946.erdos_946.variants.hildebrand_lower_bound Improved lower bound in [Hi85]: $Ω(x / (\log \log x)^3)$
Erdos946.erdos_946.variants.upper_bound Upper bound in [EPS87]: $O(x / \sqrt{\log \log x})$
Erdos949.erdos_949 Let $S \subseteq \mathbb{R}$ be a set containing no solutions to $a + b = c$
Erdos949.erdos_949.variants.sidon Let $S\sub \mathbb{R}$ be a Sidon set
Erdos95.erdos_95.variants.guth_katz Guth and Katz [GuKa15] proved the upper bound $\sum_i f(u_i)^2 \ll n^3\log n$
Erdos950.f Let $f(n) = \sum_{p<n}\frac{1}{n-p}$
Erdos950.erdos_950.parts.i Is it true that $\liminf f(n)=1$?
Erdos950.erdos_950.parts.ii Is it true that $\limsup f(n)=\infty$?
Erdos950.erdos_950.parts.iii Is it true that $f(n)=o(\log\log n)$ for all $n$?
Erdos950.erdos_950.variants.debruijn_erdos_turan This function was considered by de Bruijn, Erdős, and Turán, who showed that $\sum_{n<x}f(n)\sim \sum_{n<x}f(n)^2\sim x$
Erdos950.erdos_950.variants.weaker_pi Documented declaration: weaker_pi
Erdos950.erdos_950.variants.weaker_pi_implies_f He notes that if $\pi(x)< \pi(y)+O\left(\frac{x-y}{\log x}\right)$ for all $y0$ then $f(n)\ll \log\log\log n$
Erdos950.erdos_950.variants.sum_primes The study of $f(p)$ is even harder, and Erdős could not prove that $\sum_{p<x}f(p)^2\sim \pi(x)$
Erdos951.Erdos951Prop Documented declaration: Erdos951Prop
Erdos951.erdos_951.variants.isBeurlingPrimes If a strictly increasing sequence a : ℕ → ℝ has property Erdos951Prop and 1 < a 0, then a is a sequence of Beurling prime numbers
Erdos951.erdos_951 If 1 < a 0 < ... has property Erdos951Prop, is it true that #{a i ≤ x} ≤ π x?
Erdos951.erdos_951.variants.beurling Documented declaration: beurling
Erdos952.erdos_952 Is there an infinite sequence of distinct Gaussian primes $x_1,x_2,\ldots$ such that $\lvert x_{n+1}-x_n\rvert \ll 1$?
Erdos955.s Let $s(n)=\sigma(n)-n=\sum_{\substack{d\mid n\\ d<n}}d$ be the sum of proper divisors function
Erdos955.erdos_955 If $A\subset \mathbb{N}$ has density $0$ then $s^{-1}(A)$ must also have density $0$
Erdos955.erdos_955.variants.positive_density It is possible for $s(A)$ to have positive density even if $A$ has zero density (for example taking $A$ to be the product of two distinct primes)
Erdos955.erdos_955.variants.empty_preimage Erdős [Er73b] proved that there are sets $A$ of positive density such that $s^{-1}(A)$ is empty
Erdos955.erdos_955.variants.pollack_primes Pollack [Po14b] has shown that this is true if $A$ is the set of primes
Erdos955.erdos_955.variants.troupe_unusually_many_prime_factors Troupe [Tr15] has shown that this is true if $A$ is the set of integers with unusually many prime factors
Erdos955.erdos_955.variants.troupe_sum_of_two_squares Troupe [Tr20] has also shown this is true if $A$ is the set of integers which are the sum of two squares
Erdos955.erdos_955.variants.pollack_pomerance_thompson_bound Documented declaration: pollack_pomerance_thompson_bound
Erdos958.IsEquidistantOnLine A is a set of equidistant points on a line: there are a point a and a non-zero direction v such that A consists of the points a + i • v for 0 ≤ i < #A
Erdos958.IsEquidistantOnCircle Documented declaration: IsEquidistantOnCircle
Erdos958.erdos_958 Let $A\subset \mathbb{R}^2$ be a finite set of size $n$, and let $\{d_1,\ldots,d_k\}$ be the set of distances determined by $A$
Erdos959.indexPairs The unordered index pairs, represented uniquely by the orientation i < j
Erdos959.distanceValues The finite set of distinct distances determined by a configuration
Erdos959.frequency $f(d)$: the number of unordered pairs in $P$ determining distance $d$
Erdos959.runnerUpFrequency For a fixed distance $d$, the largest multiplicity among all other represented distances (Finset.sup returns $0$ when there is no other value)
Erdos959.multiplicityGap $f(d_1) - f(d_2)$: the gap between the largest and second-largest distance multiplicities
Erdos959.Admissible A configuration represents a set of size $n$ (injective) and has at least two distinct distances, so that $d_2$ exists
Erdos959.AttainableGap A natural number occurs as the top-two multiplicity gap of some admissible $n$-point configuration
Erdos959.extremalGap The extremal quantity of the problem, $\max_{|A| = n} (f(d_1) - f(d_2))$
Erdos959.erdos_959 Let $A\subseteq \mathbb{R}^2$ be a set of size $n$ and let $\{d_1,\ldots,d_k\}$ be the set of distinct distances determined by $A$
Erdos959.erdos_959.lower_bound Documented declaration: lower_bound
Erdos96.convexUnitDistanceCounts The set of all possible numbers of unit distances determined by the vertices of a convex $n$-gon
Erdos96.convexUnitDistanceCounts_bddAbove This lemma confirms that the set of possible unit-distance counts is bounded above, which ensures that taking the supremum (sSup) is a well-defined operation
Erdos96.maxConvexUnitDistances The maximum number of unit distances determined by the vertices of a convex $n$-gon
Erdos96.erdos_96 If $n$ points in $\mathbb{R}^2$ form a convex polygon then there are $O(n)$ many pairs which are distance $1$ apart
Erdos961.erdos_961.sylvester_schur Sylvester and Schur [Er34] proved that every set of $k$ consecutive integers greater than $k$ contains an integer divisible by a prime greater than $k$, i.e. not $(k+1)$-smooth
Erdos961.erdos_961.variants.well_defined There exists $n$ such that Erdos961Prop k n holds
Erdos961.f For $k$, let $f(k)$ be the minimal $n$ such that every set of $n$ consecutive integers $>k$ contains an integer divisible by a prime $>k$, i.e. not $(k+1)$-smooth
Erdos961.erdos_961 It is conjectured that $f(k) \ll (\log k)^O(1)$
Erdos961.erdos_961.variants.erdos_upper_bound Erdos [Er55d] proved $f(k) < 3 \frac{k}{\log k}$ for sufficiently large $k$
Erdos961.erdos_961.variants.jutila_ramachandra_shorey_upper_bound Jutila [Ju74], and Ramachandra--Shorey [RaSh73] proved a stronger upper bound $f(k) \ll \frac{\log \log \log k}{\log \log k} \frac{k}{\log k}$
Erdos962.Erdos962Prop Erdos962Prop n k : there exists $m \le n$ such that each of $m+1, \ldots, m+k$ has a prime divisor strictly larger than $k$
Erdos962.k Let $k(n)$ be the maximal $k$ such that there exists $m \le n$ with $m+1, \ldots, m+k$ each divisible by a prime $> k$
Erdos962.erdos_962 Main conjecture: $\log k(n) \le (\log n)^{(1/2 + o(1))}$
Erdos962.erdos_962.variants.tang_lower_bound Tang's lower bound [Tang]: $\log k(n) \ge (1/\sqrt{2} - o(1)) * \sqrt{\log n * \log \log n}$
Erdos962.erdos_962.variants.tao_upper_bound Tao's upper bound [Tao]: $k(n) \le (1 + o(1)) * n^{1/2}$
Erdos965.erdos_965.variants.generalization For every 2-coloring of ℝ, is there some $k ≥ 2$ and an uncountable set $A ⊆ ℝ$ such that all sums $a_1 + \cdots + a_k$ of $k$ distinct elements of $A$ have the same colour?
Erdos966.erdos_966 Let $k,r\geq 2$
Erdos967.summand The summand $\frac{1}{n^{1+it}}$, for a natural number $n$ and $t\in\mathbb{R}$
Erdos967.erdos_967.variants.yip Yip [Yi25] has proved that this is not always true - in fact, for any real
Erdos967.erdos_967.variants.finite It remains open whether this is true for every finite sequence of integers
Erdos967.erdos_967.variants.two_three_five A question of Erdős and Ingham [ErIn64]
Erdos967.erdos_967.variants.tauberian Their interest in this problem arose from their proof that the statement that there are no such zeros is equivalent to the fact that, for any non-decreasing
Erdos968.u u n is the normalized nth prime, defined as pₙ / (n+1) where pₙ is the nth prime (with 0.nth Nat.Prime = 2)
Erdos968.erdos_968 Does the set {n | u n < u (n+1)} have positive lower density?
Erdos968.erdos_968.variants.sum_abs_diff_isTheta_log_sq Erdős and Prachar proved ∑_{pₙ < x} |u (n+1) - u n| ≍ (log x)^2 (see [ErPr61])
Erdos968.erdos_968.variants.decreasing_steps_pos_lower_density Erdős and Prachar proved that the set {n | u n > u (n+1)} has positive lower density (see [ErPr61])
Erdos968.erdos_968.variants.infinite_increasingTriples Erdős asked whether there are infinitely many solutions to uₙ < uₙ₊₁ < uₙ₊₂
Erdos968.erdos_968.variants.infinite_decreasingTriples Erdős asked whether there are infinitely many solutions to uₙ > uₙ₊₁ > uₙ₊₂
Erdos97.HasNEquidistantPointsAt A set of points $A$ has n equidistant points at $p$ if there exist at least $n$ other points in $A$ that are equidistant from $p$
Erdos97.HasNEquidistantPointsOn A set of points $A$ has n equidistant points on a set of points $B$ if for every point in $B$, there exist at least $n$ other points in $A$ that are equidistant from it
Erdos97.HasNEquidistantProperty A set of points $A$ has n equidistant property if for every point in $A$, there exist at least $n$ other points in $A$ that are equidistant from it
Erdos97.HasNUnitDistancePointsAt A set of points $A$ has n unit distance points at $p$ if there exist at least $n$ other points in $A$ that are at unit distance from $p$
Erdos97.HasNUnitDistancePointsOn A set of points $A$ has n unit distance points on a set of points $B$ if for every point in $B$, there exist at least $n$ other points in $A$ that are at unit distance from it
Erdos97.HasNUnitDistanceProperty A set of points $A$ has n unit distance property if for every point in $A$, there exist at least $n$ other points in $A$ that are at unit distance from it
Erdos97.erdos_97 Does every convex polygon have a vertex with no other 4 vertices equidistant from it?
Erdos97.erdos_97.variants.three_equidistant Documented declaration: three_equidistant
Erdos97.erdos_97.variants.k_equidistant Erdős also conjectured that there is a $k$ for which every convex polygon has a vertex with no other $k$ vertices equidistant from it
Erdos97.erdos_97.variants.three_unit_distance Fishburn and Reeds [FiRe92] have found a convex polygon on 20 points such that every vertex has three vertices equidistant from it (and this distance is the same for all vertices)
Erdos97.IsCut A two-part partition $\{A, B\}$ of $V$ is a cut if the convex hulls of $A$ and $B$ are disjoint
Erdos97.erdos_97.variants.three_unit_distance_cut_min Fishburn and Reeds [FiRe92] also proved that the smallest
Erdos970.IsJacobsthalBound IsJacobsthalBound k m says that every interval of m consecutive integers contains an integer coprime to every positive natural number having at most k distinct prime factors
Erdos970.jacobsthalFunction Jacobsthal's function, uniformly parametrized by the maximum number of distinct prime factors
Erdos970.erdos_970 Documented declaration: erdos_970
Erdos971.leastCongruentPrime leastCongruentPrime a d is the least prime congruent to a modulo d
Erdos971.erdos_971 Let p(a, d) be the least prime congruent to a (mod d)
Erdos971.erdos_971.variants.infinite_sequence Erdős [Er49c] proved that the statement in erdos_971 holds for infinitely many values of d
Erdos971.erdos_971.variants.many_small Erdős [Er49c] proved that for any ε > 0 we have p(a, d) < ε * φ(d) * log d for ≫_ε φ(d) many values of a (for all large d)
Erdos972.primeSet The set of primes p such that Nat.floor (α * p) is also prime
Erdos972.erdos_972 Erdős problem 972
Erdos973.erdos_973 Does there exist a constant
Erdos973.erdos_973.variants.luo_yang_zhu The result of [LYZ26] that settles the problem: for every fixed
Erdos973.erdos_973.variants.le_one Erdős proved (as described on p.35 of [Tu84b]) that such a sequence does exist with $\lvert z_i\rvert\leq 1$
Erdos973.erdos_973.variants.m2_bounds Documented declaration: m2_bounds
Erdos973.erdos_973.variants.tang Documented declaration: tang
Erdos974.IsTuranConfiguration Documented declaration: IsTuranConfiguration
Erdos974.erdos_974 Let $z_1,\ldots,z_n\in \mathbb{C}$ be a sequence such that $z_1=1$
Erdos974.erdos_974.variants.two_tuples Erdős speculates that this may be true if there are two distinct $(n-1)$-tuples of consecutive values of $s_k$ which are $0$
Erdos975.Erdos975Sum Sum of $\tau(f(n))$ from 0 to ⌊x⌋ for a polynomial $f \in \mathbb{Z}[X]$
Erdos975.erdos_975 For an irreducible polynomial
Erdos975.erdos_975.variants.upper_bound The correctness of the growth rate is shown in [Va39] (lower bound) and [Er52b] (upper bound)
Erdos975.erdos_975.variants.lower_bound Lower bound for the growth rate of Erdos975Sum, shown in [Va39]
Erdos975.erdos_975.variants.quadratic When $f$ is an irreducible quadratic polynomial, the question is answered first by Hooley [Ho63]
Erdos975.erdos_975.variants.n2_plus_1_strong More concrete example for $f(n) = n^2 + 1$, where the asymptote is $\sum_{n \le x} \tau(n^2 + 1) \sim \frac{3}{\pi} x \log x + O(x)$
Erdos975.erdos_975.variants.n2_plus_1 Asymptotics for Erdos975Sum with $f(X) = X^2 + 1$
Erdos977.erdos_977.variants.schinzel Schinzel [Sc62] proved that $P(2^n-1)>2n$ for $n>12$
Erdos977.erdos_977.variants.stewart Stewart [St13] proved that $P(2^n-1)\gg n^{1+\frac{1}{104\log\log n}}$ for all large $n$
Erdos977.erdos_977.variants.factorial In [Er65b] Erdős also asks about $P(n!+1)$: is it true that $P(n!+1)/n\to\infty$?
Erdos977.erdos_977.variants.lai The best-known unconditional result, due to Lai [La21], is that $$\limsup \frac{P(n!+1)}{n} \geq 1+9\log 2\approx 7.238.$$
Erdos978.erdos_978.variants.sub_one Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient
Erdos978.erdos_978.parts.i Let f ∈ ℤ[X] be an irreducible polynomial with positive leading coefficient
Erdos978.erdos_978.variants.sub_two Documented declaration: sub_two
Erdos978.erdos_978.parts.ii If $k>3$ (and $k \neq 2^l$), and for all primes $p$ there exists $n$ such that $p^{k-2}\nmid f(n)$, then are there infinitely many $n$ for which $f(n)$ is $(k-2)$-power-free?
Erdos978.erdos_978.parts.iii Does n ^ 4 + 2 represent infinitely many squarefree numbers?
Erdos979.erdos_979 Let $k ≥ 2$, and let $f_k(n)$ count the number of solutions to $n = p_1^k + \dots + p_k^k$, where the $p_i$ are prime numbers
Erdos979.erdos_979.variants.k2 Erdős [Er37b] proved that if $f_2(n)$ counts the number of solutions to $n = p_1^2 + p_2^2$, where $p_1$ and $p_2$ are prime numbers, then $\limsup f_2(n) = \infty$
Erdos98.h $h(n)$ is the minimum number of distinct distances determined by any $n$-point set in $\mathbb{R}^2$ in general position (no three collinear, no four cocyclic)
Erdos98.erdos_98 Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances
Erdos98.erdos_98.variants.upper_bound Erdős could not even prove $h(n)\geq n$
Erdos982.erdos_982 If $n$ distinct points in $\mathbb{R}^2$ form a convex polygon then some vertex has at least $\lfloor\frac{n}{2}\rfloor$ different distances to other vertices
Erdos985.erdos_985 Is it true that, for every prime $p$, there is a prime $q \leq p$ which is a primitive root modulo $p$?
Erdos985.erdos_985.variants.two_three_five_primitive_root Heath-Brown proved that at least one of 2, 3, or 5 is a primitive root for infinitely many primes $p$
Erdos987.A For an infinite sequence $x_1, x_2, \ldots \in (0, 1)$, define $$A_k = \limsup_{n \to \infty} \left\lvert \sum_{j \le n} e(k x_j) \right\rvert,$$ where $e(x) = e^{2\pi i x}$
Erdos99.HasMinDist1 A set has minimum distance $1$ if all pairwise distances are at least $1$, and the minimum is achieved
Erdos99.FormsEquilateralTriangle Three points form an equilateral triangle of side length 1
Erdos99.erdos_99 For sufficiently large n, is it the case that any set of n points with minimum distance $1$ that minimizes diameter must contain an equilateral triangle of side length 1?
Erdos990.rootArgCount The number of roots of f, counted with multiplicity, whose argument lies in I
Erdos990.M For $f = a_0 + \cdots + a_dx^d$, the quantity $M=\frac{\lvert a_0\rvert+\cdots +\lvert a_d\rvert}{(\lvert a_0\rvert\lvert a_d\rvert)^{1/2}}$
Erdos990.erdos_990 Let $f=a_0+\cdots+a_dx^d\in \mathbb{C}[x]$ be a polynomial
Erdos990.erdos_990.variants.erdos_turan Erdős and Turán [ErTu50] proved such an upper bound with $n$ replaced by $d$
Erdos990.erdos_990.variants.hayman Hayman [Ha72b] proved
Erdos990.erdos_990.variants.counterexample Documented declaration: counterexample
Erdos994.visitAverage The proportion $\frac{1}{n}\sum_{1\leq k\leq n}1_{\{k\alpha \}\in E}$ of the fractional parts $\{k\alpha\}$, $1\leq k\leq n$, which lie in E
Erdos994.erdos_994 Let $E\subseteq (0,1)$ be a meaurable subset with Lebesgue measure $\lambda(E)$
Erdos994.erdos_994.variants.simultaneous Read literally, with "for almost all
Erdos995.partialSum The partial sum $S_N(\alpha) = \sum_{k < N} f(\{\alpha n_k\})$
Erdos995.erdos_995 Erdős Problem 995: For every lacunary sequence
Erdos995.erdos_995.variants.erdos_lower_bound Erdős [Er49] showed the growth cannot be pushed below
Erdos996.erdos_996 Documented declaration: erdos_996
Erdos996.erdos_996.variants.log2 The following theorem is proved in [Ma66]
Erdos997.IsWellDistributed Call
Erdos997.erdos_997.variants.lacunary Erdős proved that, if $n_k$ is a lacunary sequence, then the sequence $\{ \alpha n_k\}$ is not well-distributed for almost all $\alpha$
Erdos997.erdos_997.variants.irrational He also claimed in [Er64b] to have proved that there exists an irrational $\alpha$ for which $\{\alpha p_n\}$ is not well-distributed
Erdos999.IsApproximable x is approximable with respect to f if $$\left\lvert x-\frac{p}{q}\right\rvert < \frac{f(q)}{q}$$ has infinitely many solutions with $(p,q)=1$
Erdos999.erdos_999 For any function
Erdos999.erdos_999.variants.easy_direction It is easy to prove that divergence of the series follows from approximability almost everywhere (the Borel–Cantelli lemma)
Erdos999.erdos_999.variants.erdos Erdős [Er64b] proved the conjecture in the special case when $f(q)q$ is bounded
Green1.green_1 Let $A$ be a set of $n$ positive integers
Green12.green_12 Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$
Green14.mixedMonoAPGuaranteeSet The set of natural numbers $N$ such that any 2-coloring of ${1, ..., N}$ contains a monochromatic arithmetic progression of length $k$ (color 0) or length $r$ (color 1)
Green14.W We define the 2-colour van der Waerden numbers
Green14.green_14_polynomial Is $W(k, r)$ a polynomial in $r$, for fixed $k$?
Green14.green_14_polynomial_k_eq_3 We know $W(3, r)$ does not have polynomial growth in $r$ [Gr21, p.3]
Green14.green_14_quadratic Is $W(3, r) \ll r^2$?
Green14.green_14_lower_bound_green [Gr21] proved a lower bound of shape $W(3, r) \gg \exp(c(\log r)^{4/3-o(1)})$
Green14.green_14_lower_bound_hunter [Hu22] improved this to $W(3, r) \gg \exp(c(\log r)^{2-o(1)})$
Green14.green_14_lower_bound_brown_landman_robertson [BLR08] proved $W(3, r) \gg r^{2 - 1/\log \log r}$
Green14.green_14_lower_bound_li_shu [LiSh10] proved $W(3, r) \gg (r / \log r)^2$
Green14.green_14_upper_bound_schoen [Sc20] proves the upper bound $W(3, r) < \exp(r^{1-c})$ for some $c > 0$
Green14.green_14_upper_bound_kelley_meka [KeMe23] gives a corresponding upper bound $W(3, r) \ll \exp(C(\log r)^C)$
Green14.green_14_variant_2r2 It remains an interesting open problem to actually write down a colouring showing (say) $W(3, r) \ge 2r^2$ for some $r$
Green14.W_3_6 $W(3, 6) = 32$ from [AKS14]
Green14.W_3_7 $W(3, 7) = 46$ from [AKS14]
Green14.W_3_8 $W(3, 8) = 58$ from [AKS14]
Green14.W_3_9 $W(3, 9) = 77$ from [AKS14]
Green14.W_3_10 $W(3, 10) = 97$ from [AKS14]
Green14.W_3_11 $W(3, 11) = 114$ from [AKS14]
Green14.W_3_12 $W(3, 12) = 135$ from [AKS14]
Green14.W_3_13 $W(3, 13) = 160$ from [AKS14]
Green14.W_3_14 $W(3, 14) = 186$ from [AKS14]
Green14.W_3_15 $W(3, 15) = 218$ from [AKS14]
Green14.W_3_16 $W(3, 16) = 238$ from [AKS14]
Green14.W_3_17 $W(3, 17) = 279$ from [AKS14]
Green14.W_3_18 $W(3, 18) = 312$ from [AKS14]
Green14.W_3_19 $W(3, 19) = 349$ from [AKS14]
Green14.W_3_20 $W(3, 20) = 389$, conjectured in [AKS14, Table 2] and established by Kouril [Ko15] using FPGA-based SAT solving
Green14.W_3_21_lower $W(3, 21) \ge 416$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_21_eq [AKS14, Table 2] conjectures that $W(3, 21) = 416$
Green14.W_3_22_lower $W(3, 22) \ge 464$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_22_eq [AKS14, Table 2] conjectures that $W(3, 22) = 464$
Green14.W_3_23_lower $W(3, 23) \ge 516$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_23_eq [AKS14, Table 2] conjectures that $W(3, 23) = 516$
Green14.W_3_24_lower $W(3, 24) \ge 593$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_24_eq [AKS14, Table 2] conjectures that $W(3, 24) = 593$
Green14.W_3_25_lower $W(3, 25) \ge 656$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_25_eq [AKS14, Table 2] conjectures that $W(3, 25) = 656$
Green14.W_3_26_lower $W(3, 26) \ge 727$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_26_eq [AKS14, Table 2] conjectures that $W(3, 26) = 727$
Green14.W_3_27_lower $W(3, 27) \ge 770$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_27_eq [AKS14, Table 2] conjectures that $W(3, 27) = 770$
Green14.W_3_28_lower $W(3, 28) \ge 827$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_28_eq [AKS14, Table 2] conjectures that $W(3, 28) = 827$
Green14.W_3_29_lower $W(3, 29) \ge 868$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_29_eq [AKS14, Table 2] conjectures that $W(3, 29) = 868$
Green14.W_3_30_lower $W(3, 30) \ge 903$ from [AKS14, Table 2], proved by an explicit good partition
Green14.W_3_30_eq [AKS14, Table 2] conjectures that $W(3, 30) = 903$
Green14.W_3_31_lower $W(3, 31) > 930$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_32_lower $W(3, 32) > 1006$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_33_lower $W(3, 33) > 1063$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_34_lower $W(3, 34) > 1143$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_35_lower $W(3, 35) > 1204$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_36_lower $W(3, 36) > 1257$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_37_lower $W(3, 37) > 1338$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_38_lower $W(3, 38) > 1378$ from [AKS14, Table 3], proved by an explicit good partition
Green14.W_3_39_lower $W(3, 39) > 1418$ from [AKS14, Table 3], proved by an explicit good partition
Green15.green_15 Does there exist a Lipschitz function $f : \mathbb{N} \to \mathbb{Z}$ whose graph $\Gamma = \{(n, f(n)) : n \in \mathbb{N}\} \subseteq \mathbb{Z}^2$ is free of 3-term progressions?
Green15.green_15_ap4 The answer is YES for 4-term progressions [BJP14]
Green16.SolutionFree A set has no solution to $x + 3y = 2z + 2w$ in distinct elements
Green16.f The maximum size of a solution-free subset of $[N]$
Green16.green_16 What is the largest subset of $[N]$ with no solution to $x + 3y = 2z + 2w$ in distinct integers $x, y, z, w$?
Green16.green_16_lower_bound From [Ruzsa] $f(N) \gg N^{1/2}$
Green16.green_16_upper_bound From [Schoen and Sisask] $f(N) \ll N \cdot e^{-c(\log N)^{1/7}}$
Green16.IsTrivialZhaoSolution Documented declaration: IsTrivialZhaoSolution
Green16.ZhaoSolutionFree A set has no nontrivial solution to $x + 2y + 3z = x' + 2y' + 3z'$
Green16.g The maximum size of a Zhao-solution-free subset of $[N]$
Green16.zhao_question From [Yufei Zhao]: Is there a subset of $\{1, \ldots, N\}$ of size $N^{1/3 - o(1)}$ with no nontrivial solutions to $x + 2y + 3z = x' + 2y' + 3z'$?
Green18.numNaiveCorners The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (gx, y), (x, gy)$ are all in $A$
Green18.green_18 Suppose that $G$ is a finite group, and let $A \subset G \times G$ be a subset of density $\alpha$
Green18.numBmzCorners The number of triples $(x, y, g)$ in $G^3$ such that $g \neq e$, and $(x, y), (xg, y), (x, gy)$ are all in $A$
Green18.green_18.bmz_corners [So13] proved this is true for "BMZ corners"
Green19.IsCorner A corner in $A$ with common difference $d$ [FSS20]
Green19.S From [FSS20]: given $A \subseteq G \times G$ and $d \in G$, let $$S_d(A) = \lbrace (x, y) \in G \times G : (x, y), (x + d, y), (x, y + d) \in A \rbrace$$
Green19.ValidExponent True if the given exponent satisfies Green's conditions [Gr26]
Green19.C The infimum of all valid exponents [Gr26]
Green19.green_19 What is $C$, the infimum of all exponents $c$ for which the following is true, uniformly for $0 < \alpha < 1$?
Green19.green_19.lower [Ma21] showed that $3.13 \leq C$
Green19.green_19.upper [Ma21] showed that $C \leq 4$
Green2.maxRestrictedSumAvoidingSubsetSize We define the construction from [Sa21, p1] as $M(A) := \max \{|S| : S \subseteq A \text{ and } (S \hat{+} S) \cap A = \varnothing \}$
Green2.green_2 Let $A \subset \mathbf{Z}$ be a set of $n$ integers
Green2.green_2_lower_bound_sanders From [Sa21] it is known that there is always such an S with $|S| \gt (\log |A|)^{1+c}$
Green2.green_2_upper_bound_erdos From [Er65] it is known that $M(A) \le \frac{1}{3}|A| + O(1)$
Green2.green_2_upper_bound_choi From [Ch71] it is known that $M(A) \le |A|^{2/5 + o(1)}$
Green2.green_2_upper_bound_ruzsa From [Ru05] the best-known upper bound is $|S| \lt e^{C \sqrt{\log |A|}}$
Green21.RadoCondition The coefficients $a_1, \dots, a_k$ satisfy Rado's condition if $\sum_{i \in I} a_i = 0$ for some non-empty $I \subseteq [k]$
Green21.minColours $c(a_1, \dots, a_k)$, the least number of colours required in order to colour $\mathbb{N}$ so that there is no monochromatic solution to $a_1x_1 + \cdots + a_kx_k = 0$
Green21.green_21 Suppose that $a_1, \dots, a_k$ are integers which do not satisfy Rado's condition: thus if $\sum_{i \in I} a_i = 0$ then $I = \emptyset$
Green21.green_21.variants.fox_kleitman The answer was shown to be affirmative for $k = 3$ by Fox and Kleitman [FoKl06], who showed that $c(a_1, a_2, a_3) \leq 24$
Green21.green_21.variants.fox_kleitman_sharp Green [Gr24] is not sure that the constant $24$ of [FoKl06] is sharp, and remarks that it might be interesting to determine the sharp constant
Green21.green_21.variants.fox_kleitman_modular A question [FoKl06, Conjecture 5] of Fox and Kleitman, which they call a 'modular analogue' of Rado's Boundedness Conjecture
Green21.maxDepth The largest $d \leq r$ such that $\sum_{i \in I} a_i \equiv 0 \pmod{2^d}$ for some non-empty subset $I \subseteq [k]$, where $a_1, \dots, a_k \in \mathbb{Z}/2^r\mathbb{Z}$
Green21.green_21.variants.milicevic Milićević [ElJo23, Conjecture 11.1] conjectures the following 2-adic variant
Green21.not_radoCondition_iff Not satisfying RadoCondition is Green's phrasing "if $\sum_{i \in I} a_i = 0$ then $I = \emptyset$"
Green21.ne_zero_of_not_radoCondition A tuple failing RadoCondition has no zero coefficient, since a singleton is a non-empty subset
Green21.minColours_eq_zero In the degenerate case $k = 0$ the equation reads $0 = 0$, so the empty tuple is a monochromatic solution for every colouring and minColours takes its junk value 0
Green21.minColours_ones A case where the infimum defining minColours is a genuine minimum rather than the junk value of sInf ∅
Green22.HasMonochromaticSumProduct The monochromatic sum-product property: a colouring
Green22.N₀ $N_0(r)$ is the smallest $N$ such that every $r$-colouring of $\{1, \ldots, N\}$ has the monochromatic sum-product property
Green22.GreenSawhneyBound The upper bound function from [GrSa25]
Green22.green_22 If $\{1, \ldots, N\}$ is $r$-coloured then, for $N \geqslant N_0(r)$, there are integers $x, y \geqslant 3$ such that $x + y, xy$ have the same colour
Green22.green_22.variants.green_sawhney_bound [GrSa25, Theorem 1.1] found a permissible upper bound
Green22.green_22.variants.moreira_infinite [Mo17, Corollary 1.5] For any finite coloring of $\mathbb{N}$ there exist (infinitely many) $x, y \in \mathbb{N}$ such that $\{xy, x + y\}$ is monochromatic
Green22.green_22.variants.lower_nine Since $x, y \geq 3$, we must have $xy \geq 9$, so $N_0(r) \geq 9$ (assuming $N_0(r)$ is well-defined, which follows from [Mo17])
Green23.green_23 Suppose that $\mathbb{N}$ is finitely coloured
Green24.max013AffineTranslates The maximum number of $\lbrace 0,1,3 \rbrace$ affine translates that a set of size $n$ can contain
Green24.green_24 If $A$ is a set of $n$ integers, what is the maximum number of affine translates of the set $\lbrace 0,1,3 \rbrace$ that $A$ can contain?
Green24.variants.upper_trivial From [Aa19] p.577: the trivial upper bound is $n^2$ (non asymptotic)
Green24.variants.gamma The asymptotic constant $\gamma$ defined in [Aa19] p.579
Green24.variants.upper_HL Asymptotic upper bound (1.2) in [Aa19]
Green24.variants.lower_HL Asymptotic lower bound (1.2) in [Aa19]
Green24.variants.conjecture Conjecture p.579 in [Aa19]: $\left(\frac{1}{3} + o(1)\right) n^2$
Green25.Property25 For which values of $k$ is the following true: whenever we partition $[N] = A_1 \cup \dots \cup A_k$, $\left|\bigcup^k_{i=1} (A_i \hat{+} A_i)\right| \geq \frac{1}{10} N$?
Green25.bestLower The best-known lower bound [ESS89]
Green25.bestUpper The best-known upper bound [ESS89]
Green25.green_25 For which values of $k$ is the following true: whenever we partition $[N] = A_1 \cup \dots \cup A_k$, $\left|\bigcup^k_{i=1} (A_i \hat{+} A_i)\right| \geq \frac{1}{10} N$?
Green25.green_25.upper We conjecture that the best-known upper bound can be lowered
Green25.green_25.lower We conjecture that the best-known lower bound can be raised
Green25.green_25.variants.lower_ess89 For $k \ll \log \log N$, this is true [Gr24]
Green25.green_25.variants.upper_ess89 For $k \gg N / \log N$, it need not be true [Gr24]
Green25.green_25.variants.upper_ess89_trivial For $k \gg N / \log N$, it need not be true [Gr24]
Green26.StandardCube The standard cube in $\mathbb{F}_p^n$ is the set of points with coordinates in $\{0, 1\}$
Green26.IsCube A cube is the image of $\lbrace 0, 1\rbrace^n$ under a linear automorphism
Green26.green_26 Let $A_1, \dots, A_{100}$ be "cubes" in $\mathbb{F}^n_3$
Green26.green_26.variants.yu25 [Yu25] has solved the original problem (with 100 replaced by 4)
Green26.green_26.variants.alm91 [ALM91] showed that if 100 is replaced by $\leq c(p) \log n$ then the result is true for $\mathbb{F}^n_p$
Green26.green_26.variants.open The analogous problem in $\mathbb{F}^n_p$ remains open
Green27.m Documented declaration: m
Green27.primesAtTop atTop restricted to prime numbers
Green27.lowerBest Best-known lower bound [Be23, Theorem 3]
Green27.upperBest Best-known upper bound [Be23, Theorem 5]
Green27.green_27.equivalent What is the size of the smallest set $A \subset \mathbb{Z} / p\mathbb{Z}$ (with at least two elements) for which no element in the sumset $A + A$ has a unique representation?
Green27.green_27.lower Propose a better lower bound along primes
Green27.green_27.upper Propose a better upper bound along primes
Green27.green_27.variants.lower_be23 We have $m(p) \geq \omega(p) \log p$ for some function $\omega(p)$ tending to infinity [Be23, Theorem 3]
Green27.green_27.variants.previous_lower Previous best-known lower bound $\log p \ll m(p)$ from [St76]
Green27.green_27.variants.previous_upper Previous best-known upper bound $m(p) \ll \sqrt{p}$ from [Be23]
Green28.IsUniformOnSupport True if a PMF on $\mathbb{Z}$ is uniformly distributed on its support
Green28.indepSum The discrete convolution of two PMFs on $\mathbb{Z}$, representing the distribution of the sum of two independent random variables
Green28.green_28 Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values, and such that $X + Y$ is uniformly distributed on its range
Green29.green_29.variant Such a conclusion is known with $|S| \gg_K |A|$ [Br13 Problem 6.5, CrSi10, Sa10]
Green3.green_3 Suppose that $A \subset [0,1]$ is open and has measure greater than $\frac{1}{3}$
Green31.F Let $F(N)$ be the largest Sidon subset of $[N]$
Green31.green_31.lower Can we improve the lower bound $N^{1/2} + O(1)$, at least for infinitely many $N$?
Green31.green_31.variants.lower_eventually Can we improve the lower bound $N^{1/2} + O(1)$, for all sufficiently large $N$?
Green31.green_31.upper Can we improve the upper bound $N^{1/2} + 0.94349 N^{1/4} + O(1)$ [HZ26], at least for infinitely many $N$?
Green31.green_31.variants.upper_eventually Can we improve the upper bound $N^{1/2} + 0.94349 N^{1/4} + O(1)$ [HZ26], for all sufficiently large $N$?
Green31.green_31.variants.upper_li69 [Li69] proved $F(n) \le n^{1/2} + n^{1/4} + O(1)$
Green31.green_31.variants.upper_bfr23 [BFR23] obtained a small improvement, getting an upper bound of $F(N) \le N^{1/2} + 0.998 N^{1/4}$ for large $N$
Green31.green_31.variants.upper_cho25 The upper bound was further improved to $N^{1/2} + 0.98183 N^{1/4} + O(1)$ [CHO25]
Green31.green_31.variants.upper_hz26 [HZ26] improved the upper bound to $N^{1/2} + \gamma_0 N^{1/4} + O(1)$ with $\gamma_0 = 0.94349\ldots < 0.9435$
Green31.green_31.variants.zmod_p It is not known whether or not there exists a Sidon subset of $\mathbb{Z}/p\mathbb{Z}$ of size $(1 + o(1))\sqrt{p}$, for all $p$ [Gr24]
Green31.IsGroupSidon A subset $S$ of an abelian group is a Sidon set of the first kind in the sense of Babai and Sós [BS85] if $a + b = c + d$ with $a, b, c, d \in S$ implies $|\{a, b, c, d\}| \le 2$
Green31.isGroupSidon_iff_isSidon The two notions agree in a group where $2a = 2b$ implies $a = b$, such as a group of odd order
Green31.subsingleton_of_isSidon_of_exponent_two In a group of exponent $2$ an IsSidon set has at most one element
Green31.isGroupSidon_klein Three points of the Klein four-group form a Sidon set in the sense of Babai and Sós
Green31.not_isGroupSidon_klein_univ The whole Klein four-group is not a Sidon set in the sense of Babai and Sós, since $(0, 0) + (1, 1) = (1, 0) + (0, 1)$
Green31.green_31.variants.abelian It is not known whether, if $G$ is an abelian group of size $n$, there always exists a Sidon subset of $G$ of size $0.01\sqrt{n}$ [Gr24]
Green31.IsBinarySidon A set of binary vectors is Sidon if its pairwise sums, taken coordinatewise in ℕ, determine the unordered pair of summands
Green31.green_31.variants.sidon_01n Another very nice old problem is whether there is a Sidon subset of $\{0, 1\}^n$ of size $N^{0.51}$, where $N = 2^n$ [Gr24]
Green31.green_31.variants.sidon_01n_clz01 The best-known upper bound for a Sidon subset of $\{0, 1\}^n$ ($N = 2^n$) is $N^{0.5753}$ [CLZ01]
Green32.HasGap A set $A$ has a gap of length $L$ if there exists $x$ such that $x, x+1, \dots, x+L-1$ are all not in $A$
Green32.hasGap_zero Any set has a gap of length 0 (vacuously true)
Green32.hasGap_empty The empty set has a gap of any length
Green32.not_hasGap_univ The full set in $\mathbb{Z}/p\mathbb{Z}$ has no gap of positive length
Green32.hasGap_concrete Concrete: $\{0\}$ in $\mathbb{Z}/5\mathbb{Z}$ has a gap of length 4 starting at 1
Green32.HasLargeGapDilate The generalized problem: for a prime
Green32.green_32 Let $p$ be a prime and let $A \subset \mathbb{Z}/p\mathbb{Z}$ be a set of size $\lfloor \sqrt{p} \rfloor$
Green32.green_32.variants.sh20_general [Sh20, Theorem 1] implies a gap of at least $\lfloor 2p/|A| - 2 \rfloor$
Green32.green_32.variants.sh20_sqrt [Sh20] has used the polynomial method to show that this is true with 100 replaced by 2 [Gr24]
Green32.green_32.variants.szemeredi_regime In the regime $\omega(p) \sim c p$, this is Szemerédi's theorem [Gr24]
Green32.green_32.variants.dirichlet_regime In the regime $\omega(p) \le c \log p$, this is basically Dirichlet's lower bound for the size of Bohr sets [Gr24]
Green32.green_32.variants.log_regime Even what happens in the regime $\omega(p) \sim 10 \log p$ is unclear [Gr24]
Green32.HasCosetHole A set $A$ has a coset hole of size $L$ if there exists a subspace $W$ and a vector $v$ such that the affine space $v + W$ has size at least $L$ and is disjoint from $A$
Green32.hasCosetHole_empty The empty set in $\mathbb{F}_2^n$ has a coset hole (using the trivial subspace)
Green32.green_32.variants.finite_field Tom Sanders' finite field variant [Gr24]
Green33.green_33 Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$?
Green33.green_33.sanity_sq_bound Trivial lower bound: if $A + A = \mathbb{Z}/q\mathbb{Z}$, then $|A|^2 \geq q$, since the sumset $A + A$ has at most $|A|^2$ elements
Green35.IsUnitIntervalDensity A nonnegative integrable function on $[0,1]$ with total integral $1$
Green35.c The infimum of $\|f \ast f\|_p$ over unit-interval densities
Green35.green_35.lower Lower bound for $c(p)$ for $1 < p \le \infty$, improving the known value $\sqrt{4/7}$ at $p = 2$ or the known value $0.64$ at $p = \infty$
Green35.green_35.upper Upper bound for $c(p)$ for $1 < p \le \infty$, improving the best-known value $0.7516$ at $p = \infty$
Green35.variants.c_2_lower Lower bound for $c(2)$ from Green's first paper ([Gr01]); the constant is sqrt(4/7) (about 0.7559)
Green35.variants.c_inf_lower Best-known lower bound for $c(\infty)$ due to Cloninger and Steinerberger ([CS17])
Green35.variants.c_inf_upper Upper bound for $c(\infty)$ due to Matolcsi and Vinuesa ([MV10]); their step function has autoconvolution supremum $1.50972\ldots$, which rescales to $0.75486\ldots$
Green35.variants.c_inf_upper_ae25 Documented declaration: c_inf_upper_ae25
Green35.variants.c_inf_upper_ggtw25 Best-known upper bound for $c(\infty)$ ([GGTW25], §6.2): a step function with autoconvolution supremum at most $1.5032$, which rescales to $0.7516$
Green35.variants.c_inf_lower_young A comparison bound from Young's inequality
Green36.SimultaneousDoubleProduct The simultaneous double product property [CKS05, 4.1]
Green36.Green36Property A variant of the simultaneous double product property, as stated in [Gr24, Problem 36]
Green36.green_36 Do the following exist, for arbitrarily large $n$?
Green36.green_36.variants.cks05 Variant using the exact simultaneous double product property from [CKS05, 4.1]
Green37.IsAPCover A contains an arithmetic progression of length k and common difference d for every d ∈ {1, …, N}
Green37.m The minimum size of a subset of ℕ that contains, for each d = 1, …, N, an arithmetic progression of length k with common difference d
Green37.green_37 Given a natural number N, what is the smallest size of a subset of ℕ that contains, for each d = 1, …, N, an arithmetic progression of length k with common difference d
Green37.green_37_asymptotic Asymptotic version: determine the asymptotic behavior of m(N, k) as N grows
Green37.green_37_theta Determine, for each k, the asymptotic equivalence class (theta) of m(N, k) as N grows
Green37.m_le_mul_add_one The interval {0, …, kN} contains, for each d = 1, …, N, the arithmetic progression {0, d, …, (k - 1)d}, so m(N, k) ≤ kN + 1
Green37.green_37_lower_bound Writing F_k(N) for m(N, k), Green conjectures that F_k(N) ≫_k N^(1 - c_k) for some sequence c_k with c_k → 0 as k → ∞
Green38.IntersectsOnlyAtZero $A - A$ intersects $\{-1, 0, 1\}^n$ only at $0$
Green38.ValidCardinalities The set of cardinalities of all subsets A where A - A intersects {-1, 0, 1}^n only at 0
Green38.green_38.test_n1_lower {0, 2, 4} is a valid independent set in C_7, giving cardinality 3
Green38.LargestAdmissibleCardinality The largest subset $A \subset \mathbb{F}_7^n$ for which $A - A$ intersects $\{-1, 0, 1\}^n$ only at $0$
Green38.C₁ The lower bound constant $C_1 \approx 3.2578$ from [Po20]
Green38.C₂ The upper bound constant $C_2 \approx 3.3177$ from [La79]
Green38.green_38.lower Can we improve the lower bound?
Green38.green_38.upper Can we improve the best upper bound?
Green38.green_38.variants.best_lower The lower bound $(C_1 - o(1))^n \leqslant |A|$ where $C_1 = 367^{1/5} \approx 3.2578$, from a valid set of size $367$ in $\mathbb{F}_7^5$ [Po20, Section 9.1]
Green38.green_38.variants.lower_itty_rosin_carstensen_reichman Itty, Rosin, Carstensen and Reichman [IRCR26] found a valid set of size $134753$ in $\mathbb{F}_7^{10}$
Green38.green_38.variants.best_upper The current best upper bound is $|A| \leqslant (C_2 + o(1))^n$ where $C_2 = \frac{7 \cos(\pi/7)}{1 + \cos(\pi/7)} \approx 3.3177$ [La79, Corollary 5]
Green38.green_38.test_zero_mem_validCardinalities 0 is a valid cardinality, since the empty set vacuously satisfies the condition
Green38.green_38.test_bound_above The set of valid cardinalities we take the supremum over is bounded above
Green39.proportionCoverable The proportion of subsets of $\mathbb{Z}/p\mathbb{Z}$ of size $k$ that can cover $\mathbb{Z}/p\mathbb{Z}$ using at most $c$ translates
Green39.green_39 If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random, $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translates of $A$?
Green39.green_39.variant_101 "I do not know how to answer this even with 100 replaced by 1.01." [Gr24]"
Green39.green_39.variant_theta Similar questions are interesting with $\sqrt{p}$ replaced by $p^\theta$ for any $\theta \le 1/2$
Green4.ProdFree A set in a monoid is product-free if there are no elements x, y, z in the set such that x * y = z
Green4.green_4 What is the largest product-free set in the alternating group $A_n$?
Green4.extremalFamily Defines a family of subsets of $A_n$ where each permutation $\pi$ in a subset obeys $\pi(x)$ and $\forall v \in I$, $\pi(v)\notin I$ for a fixed $x$ and $I$
Green4.large_green_4 In the case of large n, the problem was solved in On the largest product-free subsets of the alternating groups
Green40.hammingBall The Hamming ball of radius $r$ in $\mathbb{F}_2^n$
Green40.IsCoveringSubspace $V$ is a covering subspace of $\mathbb{F}_2^n$ by $H(r)$ if $V + H(r) = \mathbb{F}_2^n$
Green40.minDensity The minimal covering density over all covering subspaces for a given n and r
Green40.f Documented declaration: f
Green40.green_40 Does $f(r) \to \infty$?
Green40.green_40.sanity_f_one The only value known is $f(1) = 1$, which follows from the existence of the Hamming code [Gr24]
Green40.green_40.upper_bound $f(r) \le r^r / r! \sim e^r$ [Gr24]
Green40.green_40.f_eq_one_for_all The possibility that f(r) = 1 for all r has not been ruled out [Gr24]
Green40.green_40.f_two_eq_one It is not known whether f(2) = 1 [Gr24]
Green40.green_40.upper_bound_f_two The best-known upper bound for $f(2)$ is $1.4238$ [CHL97]
Green40.green_40.variants.arbitrary_subsets Does $\tilde{f}(r) \to \infty$?
Green40.green_40.variants.arbitrary_subsets_sanity_f_tilde_two It is known that $\tilde{f}(2) = 1$ [St94]
Green40.green_40.f_tilde_le_f We evidently have $\tilde{f}(r) \le f(r)$ [Gr24]
Green40.green_40.variants.all_n Does $f_{\text{all}}(r) \to \infty$?
Green41.pyjamaSet The pyjama set is the set of points in the complex plane whose real part is within $\varepsilon$ of an integer
Green41.coveringCopies The set of valid numbers of rotated copies of the pyjama set of width ε that cover the plane
Green41.minCopies The minimal number of rotated copies of the pyjama set of width ε needed to cover the plane
Green41.minCopies_set_nonempty [Ma15] proved that for any $\varepsilon > 0$, finitely many rotations of the pyjama set of width $\varepsilon$ cover the plane
Green41.green_41 How many rotated (about the origin) copies of the 'pyjama set' $\\{(x, y) \in \mathbb{R}^2 : \text{dist}(x, \mathbb{Z}) \leq \varepsilon\\}$ are needed to cover $\mathbb{R}^2$?
Green41.green_41.variants.double_exponential_bound Can the triple-exponential bound of [KrLe25] be improved by one exponential?
Green41.green_41.variants.polynomial_bound Is $\varepsilon^{-C}$ rotations enough?
Green41.green_41.variants.kravitz_leng [KrLe25] have established the first quantitative bound, showing via an analysis of [Ma15]'s method that $\exp\exp\exp(\varepsilon^{-C})$ rotations suffice
Green42.fHat The real-valued Fourier transform used in the Cohn--Elkies conditions
Green42.CohnElkiesAdmissible Definition 2.1 from [CoEl03]: A function is admissible if it is continuous and both the function and its Fourier transform decay sufficiently fast
Green42.SatisfiesCohnElkiesScheme The structural rules a function must satisfy to successfully pass through the Cohn-Elkies scheme and generate some valid upper bound
Green42.CohnElkiesOptimal The statement that there exists a function in dimension d satisfying the Cohn-Elkies scheme which achieves the center density bound bound
Green42.green_42 Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?
Green42.green_42.variants.dimension_one [CoEl03] proved this when $d = 1$
Green42.green_42.variants.dimension_eight [Vi17] established the case $d = 8$
Green42.green_42.variants.dimension_twenty_four In [CKM17], [Vi17] was adapted to $d = 24$
Green44.green_44 Sieve $[N]$ by removing half the residue classes mod $p_i$, for primes $2 \leqslant p_1 < p_2 < \dots < p_{1000} < N^{9/10}$
Green44.green_44.variants.less_than_sqrt The answer is affirmative if the primes are all less than $N^{1/2}$, by the large sieve
Green45.green_45 Can we pick residue classes $a_p \pmod{p}$, one for each prime $p \leq N$, such that every integer $\leq N$ lies in at least $10$ of them?
Green46.IsCoveredByResidues Given $x$ and $y$, can we cover the interval $[1, y]$ by residue classes $a_p \pmod p$ for each prime $p \le x$?
Green46.maxY The maximum $y$ for a given $x$, cast to a real number for asymptotics
Green46.bestLower Best-known lower bound [Ra38]
Green46.bestUpper Best-known upper bound [Iw78]
Green46.green_46.improve_lower We conjecture that the best-known lower bound can be improved
Green46.green_46.improve_upper We conjecture that the best-known upper bound can be improved
Green46.green_46.improve_upper_conjectured It seems very likely that we must have $y \ll x^{1+o(1)}$ [Gr24]
Green46.green_46.variants.lower Best known lower bound: $y \gg x \frac{\log x \log \log \log x}{\log \log x}$ [Ra38]
Green46.green_46.variants.upper Best known upper bound: $y \ll x^2$ [Iw78]
Green47.green_47.variants.finite_exceptions In Green and Harper's formulation [GH14, Conjecture 1.7], the quadratic containment is allowed to fail on a finite subset of $A$
Green49.green_49 Suppose that $A \subset \mathbb{F}_2^n$ is a set with $|A + A| \leq K|A|$
Green5.green_5 Which finite groups have the smallest biggest product-free sets?
Green5.green_5.variants.kedlaya Kedlaya [Ke97] observed, refining some work of Babai and Sós [BaSo85], that it follows from the classification of finite simple groups that every finite group
Green5.green_5.variants.sl_two A good model problem would be to determine the largest product-free subsets of $\mathrm{SL}_2(\mathbb{F}_p)$
Green5.green_5.variants.gowers_sl_two For the largest product-free subsets of $\mathrm{SL}_2(\mathbb{F}_p)$ of order $n$, the best-known upper bound is $O(n^{8/9})$, due to Gowers [Go08]
Green50.green_50 Let $A \subset \mathbb{F}_2^n$ be a set of density $\alpha > 0$
Green51.guaranteedMaxCosetDim The largest dimension of a coset guaranteed to be contained in $2A$ for $A \subseteq \mathbb{F}_2^n$ with density $\alpha$
Green51.green_51 Suppose that $A \subset \mathbb{F}_2^n$ is a set of density $\alpha$
Green51.green_51.lower It is known that $A + A$ must contain a coset of dimension $\gg_\alpha n$ [Gr13]
Green51.green_51.upper It is known that $A + A$ need not contain a coset of dimension $n - \sqrt{n}$ [Gr13]
Green51.green_51.one_half Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha > 1/2 - C/\sqrt{n}$
Green51.guaranteedMaxAPLength The largest length of an arithmetic progression guaranteed to be contained in $A+A$ for $A \subseteq \{1, \dots, N\}$ with density $\alpha$
Green51.green_51.lower_ap It is known that $A + A$ must contain an arithmetic progression of length $\sim \exp(c (\log N)^{1/2})$ [Gr02]
Green51.green_51.upper_ap It is known that $A + A$ need not contain an arithmetic progression of length $\sim \exp(c (\log N)^{2/3 + \varepsilon})$, for any $\varepsilon > 0$ [Ruz91]
Green52.green_52 Suppose that $A \subset \mathbb{F}_2^n$ is a set with an additive complement of size $K$
Green53.green_53 Suppose that $\mathbb{F}_2^n$ is partitioned in to sets $A_1, ..., A_K$
Green54.gaussianMeasureInf The infinite-dimensional Gaussian measure γ∞ on ℝ^ℕ, defined as the countable product of standard Gaussian measures
Green54.green_54 Documented declaration: green_54
Green54.green_54_known_case The same statement is known to be false for the sumset $2K = K + K$ instead of $10K$
Green57.tripleKernel Uniform average over pairs (x₁, x₂) in G × G, with the third variable determined by the relation x₁ + x₂ + x₃ = g
Green57.baseΦ The generating family of functions for Φ(G)
Green57.baseΦ' The generating family of functions for Φ′(G), where the third kernel depends only on x₁ + x₂
Green57.Φ The space Φ(G) of "convex combinations" of kernels from baseΦ
Green57.Φ' The restricted space Φ′(G) of "convex combinations" of kernels from baseΦ'
Green57.functional The linear functional on G → ℂ given by φ ↦ Re ∑ g, a g * φ g
Green57.supportFn The support function of a set S with respect to a linear functional a: sup_{φ ∈ S} Re ⟨a, φ⟩
Green58.green_58 Suppose $A, B ⊆ \{1, \dots, N\}$ both have size at least $N^{0.49}$
Green60.green_60 Is there an absolute constant $c > 0$ such that, whenever $A ⊆ \mathbb{N}$ is a set of squares with $|A| ≥ 2$, the sumset $A + A$ satisfies $|A + A| ≥ |A|^{1 + c}$?
Green61.green_61 Suppose that $A + A$ contains the first $n$ squares
Green62.green_62 Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$
Green64.green_64 Do there exist infinitely many primes $p$ for which $p - 2$ has an odd number of prime factors, counted with multiplicity (i.e. $\Omega(p - 2)$ is odd)?
Green64.green_64_mem_five $5$ satisfies the condition: $5$ is prime and $5 - 2 = 3$ is prime, so $\Omega(3) = 1$ is odd
Green64.green_64_mem_seven $7$ satisfies the condition: $7$ is prime and $7 - 2 = 5$ is prime, so $\Omega(5) = 1$ is odd
Green64.green_64_not_mem_eleven $11$ does not satisfy the condition: although $11$ is prime, $11 - 2 = 9 = 3 ^ 2$ has $\Omega(9) = 2$ prime factors, which is even
Green64.green_64.variants.p_sub_one The same question as green_64 but with $p - 1$ instead of $p - 2$
Green66.IsSumOfTwoSquares A natural number is a sum of two squares if it can be written as a ^ 2 + b ^ 2
Green66.green_66 Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$?
Green66.green_66.variants.trivial_bound Green notes that a well-known, almost trivial argument gives an $O(X^{1/4})$ bound on the left
Green7.green_7.variants.positive_density Does Ulam's sequence have positive density?
Green72.AllowedSet We say a subset of $[N]^2$ is allowed for some $k$ if it contains no $k$ points which lie on a common line
Green72.AllowedSetSize The maximal size of an allowed set
Green72.allowedSetSize_le By the pigeon hole principle, the size of a subset of an $N \times N$ grid such that no $k$ points lie on a line is bounded by $\leq (k - 1) * N$
Green72.NoKInLineFor The proposition that the allowed-set size for $k$ and $N$ is $(k - 1) * N$
Green72.NoKInLine Documented declaration: NoKInLine
Green72.green_72 Green's Open Problem 72 / No-three-in-line problem: For $N$ sufficiently large, is it impossible to have $2N$ points in $[N]^2$ with no three in a line?
Green72.green_72.variants.eventually Is $2N$ attained for all sufficiently large $N$?
Green72.no_three_in_line_le For $N \leq 60$, this has been verified with computers
Green72.no_k_in_line_big In [GK2025] Grebennikov and Kwan prove that the pigeonhole bound $(k - 1) N$ is attained for $k > 10 ^ {37}$ and $N \geq k$
Green77.green_77 Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?
Green82.minZeros The minimum number of zeros in $[0,1)$ of $\sum_{a \in A} \cos(2\pi a\theta)$ over all sets $A \subset \mathbb{Z}$ of size $n$
Green82.green_82 Let $A \subset \mathbb{Z}$ be a set of size $n$
Green82.green_82_lower_bound Lower bound: every set has at least $(\log \log n)^{1-o(1)}$ zeros
Green82.green_82_upper_bound Upper bound: there exists $A$ with at most $C(n \log n)^{2/3}$ zeros
Green85.green_85 Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$
Green85.green_85_loose From [Gr24] "It is quite easy to show using Cauchy-Schwarz that there must be such a rectangle with area $\gg \alpha^2 (\log 1/\alpha)^{-1}$."
Green9.r The quantity $r_k(N)$, defined as the size of the largest subset of $\{1, \dots, N\}$ without non-trivial $k$-term arithmetic progressions
Green9.green_9_i Problem 9 (i): is $r_3(N) \ll N(\log N)^{-10}$?
Green9.green_9_ii Problem 9 (ii): is $r_5(N) \ll N(\log N)^{-c}$?
Green9.green_9_iii Problem 9 (iii): is $r_4(\mathbf{F}_5^n) \ll N^{1-c}$, where $N=5^n$?
Green94.green_94_outer_measure Let A ⊂ R be a set of positive outer measure
Green94.green_94 Let A ⊂ R be a set of positive measure
Hilbert17.hilbert_17th_problem Hilbert's 17th problem: every non-negative multivariate polynomial is a sum of squares of rational functions
Hilbert17.f The statement is false in general if we restrict to polynomials
Hilbert17.f_nonneg The Motzkin polynomial is non-negative everywhere
Hilbert17.f_not_sum_of_squares The Motzkin polynomial cannot be written as a sum of squares of polynomials
Hilbert17.Hilbert17thProblemHomogenousPoly Documented declaration: Hilbert17thProblemHomogenousPoly
Hilbert17.hilbert_17th_problem_poly Hilbert's 17th problem for homogeneous polynomials: characterization of dimensions and degrees where non-negative polynomials are sums of squares of polynomials
Hilbert5.admitsLieGroupStructure_circle The circle group admits a Lie group structure
Hilbert5.admitsLieGroupStructure_multiplicative_int The discrete group ℤ admits a Lie group structure
Hilbert5.hilbert_smith_conjecture Hilbert–Smith conjecture: every locally compact topological group acting continuously and faithfully on a connected finite-dimensional topological manifold is a Lie group
Hilbert5.hilbert_smith_conjecture.variants.riemannian Documented declaration: riemannian
Hilbert5.hilbert_smith_conjecture.variants.dimension_three Pardon's theorem (2013): the Hilbert–Smith conjecture holds for connected 3-manifolds, see [Pardon 2013]
Hilbert5.hilbert_smith_padic_formulation p-adic formulation: the p-adic integers ℤ_[p] cannot act continuously and faithfully on a connected finite-dimensional topological manifold
Hilbert5.hilbert_fifth_problem Documented declaration: hilbert_fifth_problem
Kourovka.«19.25».kourovka_19_25 Let $G$ and $H$ be finite groups of the same order with $\sum_{g \in G} \phi(|g|) = \sum_{h \in H} \phi(|h|)$, where $\phi$ is the Euler totient function
Kourovka.«1.35c».IsBiInvariant A relation on a group is bi-invariant if it is preserved by both left and right multiplication
Kourovka.«1.35c».ProOrderable ProOrderable G means every bi-invariant partial order extends to a bi-invariant linear order
Kourovka.«1.35c».kourovka_1_35c Do there exist simple pro-orderable groups?
Kourovka.«1.40».engelCommutator The iterated commutator $[x,\, {}_n y]$, defined by $[x,\, {}_0 y] = x$ and $[x,\, {}_{n+1} y] = [[x,\, {}_n y], y]$
Kourovka.«1.40».IsEngelElement An element $y$ of a group $G$ is a (left) Engel element if for every $x \in G$ there is some $n$ with $[x,\, {}_n y] = 1$
Kourovka.«1.40».IsEngelGroup A nilgroup (Engel group) is a group in which every element is an Engel element
Kourovka.«1.40».kourovka_1_40 Is a group a nilgroup if it is the product of two normal nilsubgroups?
Kourovka.«1.74».IsMinimalTopologicalGroup A minimal topological group in Platonov's sense: a non-discrete Hausdorff topological group all of whose proper closed subgroups are discrete
Kourovka.«1.74».kourovka_1_74 Describe all minimal topological groups, that is, all non-discrete Hausdorff topological groups whose proper closed subgroups are all discrete
Kourovka.«1.74».IsTarskiMonster A Tarski monster group: an infinite group in which every non-trivial proper subgroup has order a fixed prime $p$
Kourovka.«1.74».kourovka_1_74.variants.tarski_monster There exists a Tarski monster group that admits a non-discrete Hausdorff group topology
Kourovka.«20.76».kourovka_20_76 Let $G$ be a finite $p$-group and assume that all abelian normal subgroups of $G$ have order at most $p^k$
Kourovka.«21.149».IsLocallyRightLinear An order automorphism $f$ of $\mathbb{R}$ is locally right $H$-linear if every point has a right neighbourhood on which $f$ is affine with slope in $H \le \mathbb{R}_{>0}$
Kourovka.«21.149».IsDlabGroup $A$ is one of the six Dlab groups with slope group $H$
Kourovka.«21.149».DlabLt Dlab's order on a group $G$ of order automorphisms of $\mathbb{R}$: $f < g$ if $f(x) < g(x)$ at some point $x$ and every point $y$ with $g(y) < f(y)$ lies to the right of $x$
Kourovka.«8.8».kourovka_8_8a Does there exist a non-cyclic finitely-generated group $G$ containing an element $a$ such that every element of $G$ is conjugate to a power of $a$?
Kourovka.«8.8».kourovka_8_8b Does there exist a non-cyclic finitely presented group $G$ which contains an element $a$ such that each element of $G$ is conjugate to some power of $a$?
LamLitt.IsSolutionOfAlgebraicODE A power series
LamLitt.IsCoeffIntegralAdjointInvNat There exists $N$ such that for all $n$, the $n$-th coefficient of $f$ is in $\mathbb{Z}[1/N]$
LamLitt.omegaIntegral For an integer-valued function
LamLitt.omegaSuperlinear The growth condition on $\omega$: the ratio $\omega(p) / p$ tends to infinity as the prime $p$ tends to infinity, i.e. $\lim_{p \to \infty} \omega(p) / p = \infty$
LamLitt.lam_litt.variants.eisenstein Eisenstein's theorem (1852): an algebraic power series over $\mathbb{Q}[z]$ has bounded denominators, i.e., there exists $N$ such that all coefficients lie in $\mathbb{Z}[1/N]$
LamLitt.den_coprime_of_mem_adjoinInvNat Every element of $\mathbb{Z}[1/N]$ has denominator coprime to any prime $p$ not in the prime factor set of $N$ (vacuously, the case $N = 0$ gives $\mathbb{Z}[1/N] = \mathbb{Z}$)
LamLitt.lam_litt.variants.integrality_implies_omega_integrality Textbook implication: integrality (2) trivially implies ω(p)-integrality (3)
LamLitt.lam_litt.variants.omega_integrality_implies_algebraicity Documented declaration: omega_integrality_implies_algebraicity
LamLitt.lam_litt.variants.integrality_implies_algebraicity 2) implies 1): if the coefficients of $f$ are in $\mathbb{Z}[1/N]$ for some $N$, then $f$ is algebraic over $\mathbb{Q}[z]$
Mathoverflow10799.μ Start with a set $X=\{1,2,...,n\}$ of $n$ elements and the family $2^X$ of all subsets of $X$
Mathoverflow10799.μ_half_eq_uniform For $p = 1/2$, the $p$-biased measure is the uniform distribution: $\mu_{1/2}(S) = (1/2)^n$ for every $S \subseteq [n]$
Mathoverflow10799.μFamily The $p$-biased measure of a family $\mathcal F \subseteq 2^{[n]}$, i.e. $\mu_p(\mathcal F) = \sum_{S \in \mathcal F} \mu_p(S)$
Mathoverflow10799.boundaryCount Given a family
Mathoverflow10799.boundaryCount_equiv Test lemma showing that boundaryCount is equivalent to counting subsets $T$ that differ from $S$ in exactly one element and exactly one of $S, T$ belongs to $F$
Mathoverflow10799.edgeBoundary The edge-boundary of $F$ is the expectation of $h(S)$ (according to $\mu_p$) over all subsets $S$ of $X$
Mathoverflow10799.IsMonotoneIncreasing A family $F$ of subsets of $2^X$ is monotone increasing if when $S$ belongs to $F$ and $T$ contains $S$ then $T$ also belongs to $F$
Mathoverflow10799.IsOptimal We say that a family is optimal for $\mu_p$ if the isoperimetric inequality (IR) is sharp up to a multiplicative constant $1000 \log (1/p)$
Mathoverflow10799.IsOptimalWith Documented declaration: IsOptimalWith
Mathoverflow10799.isOptimal_iff_isOptimalWith IsOptimal is IsOptimalWith for the constant $C = 1000$
Mathoverflow10799.mathoverflow_10799.variants.weak_kahn_kalai Weaker version proven by Kahn–Kalai (2006): for every fixed
Mathoverflow10799.discrete_isoperimetric_inequality Now a famous isoperimetric relation asserts that (IR) $I^p(F) \ge \frac{1}{p} \mu_p(F) \log_p \mu_p(F)$ This relation holds for monotone increasing families $F$ and $0 < p < 1$
Mathoverflow10799.μ_sum_eq_one The $p$-biased measure is a probability distribution: it sums to $1$ over all subsets
Mathoverflow10799.μFamily_univ The measure of the full power set is $1$
Mathoverflow10799.boundaryCount_empty The boundary count is zero for the empty family (no set is in $\mathcal F$)
Mathoverflow10799.edgeBoundary_empty The edge boundary is zero for the empty family
Mathoverflow10799.boundaryCount_univ The boundary count is zero for the full family (every set is in $\mathcal F$)
Mathoverflow10799.edgeBoundary_univ The edge boundary is zero for the full family
Mathoverflow17560.mathoverflow_17560 If $2^x$ and $3^x$ are integers, then $x$ must be an integer
Mathoverflow17560.mathoverflow_17560.variants.all_nats If for each natural number $n$ the number $n^x$ is an integer then $x$ must also be an integer
Mathoverflow17560.mathoverflow_17560.variants.with_5 If $2^x$, $3^x$ and $5^x$ are integers, then $x$ must be an integer
Mathoverflow1973.unitSphere The unit n-sphere, defined as Metric.sphere 0 1 in EuclideanSpace ℝ (Fin (n + 1))
Mathoverflow21003.mathoverflow_21003 Is there any polynomial $f(x, y) \in \mathbb{Q}[x, y]$ such that $f : \mathbb{Q} \times \mathbb{Q} \rightarrow \mathbb{Q}$ is a bijection?
Mathoverflow235893.IsConnectedMap For topological spaces $X$ and $Y$ we say a function $f : X → Y$ is connected is it sends connected sets to connected sets
Mathoverflow235893.Continuous.isConnectedMap By a standard result, every continuous map is connected
Mathoverflow235893.isConnected_iff_ordConnected_and_nonempty A set in $\mathbb{R}$ is connected if and only if it is order-connected and non-empty
Mathoverflow235893.isConnectedMap_symm_of_R If $f : \mathbb{R} \to \mathbb{R}$ is a connected bijection, then its inverse is also a connected bijection
Mathoverflow235893.isConnectedMap_comp The composition of two connected maps is a connected map
Mathoverflow235893.isConnectedMap_homeomorph A homeomorphism is a connected map
Mathoverflow235893.isConnectedMap_symm_of_E1 If $f : \mathbb{R}^1 \to \mathbb{R}^1$ is a connected bijection, then its inverse is also a connected bijection
Mathoverflow235893.mathoverflow_235893 Assume for $n>1$, $f:\mathbb{R}^n\to\mathbb{R}^n$ is a bijection, where $\mathbb{R}^n$ is equipped with the standard topology
Mathoverflow235893.mathoverflow_260589 There exists a connected bijection ℝ → ℝ^2 where the inverse is not connected, proven in mathoverflow/260589 by user Gro-Tsen
Mathoverflow31809.mathoverflow_31809 Does there exist a category that is pretriangulated but not triangulated?
Mathoverflow339137.IsZeroOne The predicate that all coefficients of a polynomial are either zero or one
Mathoverflow339137.mathoverflow_339137 Let $P(x), Q(x) ∈ ℝ[x]$ be two monic polynomials with non-negative coefficients
Mathoverflow339137.mathoverflow_339137_probabilistic Green's Open Problem 28 is the probabilistic reformulation of Mathoverflow 339137
Mathoverflow34145.Rectangle A rectangle is specified by its width, height, starting point, and rotation
Mathoverflow34145.rigidMotion A combination of a rotation and a translation to map the standard rectangle to the desired rectangle
Mathoverflow34145.scale A scaling to map the unit square to a standard rectangle
Mathoverflow34145.unitSquare The unit square
Mathoverflow34145.Rectangle.toSet Converts a rectangle to a set in ℝ × ℝ
Mathoverflow34145.lbMeasure The standard Lebesgue measure on ℝ²
Mathoverflow34145.lbMeasure_rigidMotion lbMeasure is invariant under rigidMotion start θ
Mathoverflow34145.lbMeasure_scale lbMeasure is scaled by scale
Mathoverflow34145.lbMeasure_unitSquare The Lebesgue measure of the unit square is 1
Mathoverflow34145.lbMeasure_rectangle_toSet The Lebesgue measure of the a rectangle r is r.width * r.height
Mathoverflow34145.tsum_area_eq_one The areas of the required rectangles sum to 1
Mathoverflow34145.Configuration A configuration of rectangles of sides 1 / (n + 1) and 1 / (n + 2)
Mathoverflow34145.Configuration.IsPacking A "packing" means that the interiors of any two rectangles are disjoint
Mathoverflow34145.rectangles_cover_unit_square Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?
Mathoverflow34145.rectangles_pack_unit_square Equivalently, can a unit square be packed with rectangles of width 1 / (n + 1) and height 1 / (n + 2)?
Mathoverflow34145.rectangles_pack_square_133_div_132 It is known that packing the rectangles into a square of side length 133/132 is possible
Mathoverflow34145.rectangles_pack_square_501_div_500 It is known that packing the rectangles into a square of side length 501/500 is possible
MathOverflow434111.S $S_n$ is the sum of the first n primes, i.e. $S_n=p_1+\dots+p_n$, $1$-indexed, matching the MathOverflow question
MathOverflow434111.piRestricted Documented declaration: piRestricted
MathOverflow434111.restricted_prime_number_theorem The conjecture claims that $\pi_n\sim\frac n{2\ln(n)}$
MathOverflow434111.restricted_prime_number_theorem.variants.even_subsequence Meštrović's original formulation [Me18, Conjecture 3.3]: the sequence of sums of the first
Mathoverflow486451.exists_semiring_unique_left_maximal_not_unique_right_maximal There exists a semiring with a unique left maximal ideal but more than one right maximal ideals
Mathoverflow486451.exists_semiring_unique_left_right_maximal_ne There exists a semiring with a unique left maximal ideal and a unique right maximal ideal which are not the same as sets
Mathoverflow75792.Reachable The inductively defined predicate that m is reachable in n steps
Mathoverflow75792.Reachable.decAux Auxiliary decision procedure for Reachable, taking a fuel argument f bounding m
Mathoverflow75792.complexity The (Mahler-Popken) complexity of n: the minimum number of 1s needed to express a given number using only addition and multiplication
Mathoverflow75792.Reachable.five_pow_six 5^6 = 15625 = 1 + 2^3 * 3^2 * (1 + 2^3 * 3^3)!
Mathoverflow75792.complexity_five_pow Is 5n the complexity of 5^n for 0 < n?
Mathoverflow75792.complexity_three_pow Is 3n the complexity of 3^n for 0 < n?
Mathoverflow75792.complexity_two_pow Is 2n the complexity of 2^n for 0 < n?
BSD.Weak The weak Birch and Swinnerton-Dyer conjecture for a number field
BSD.weak_birch_swinnerton_dyer_conjecture Weak Birch and Swinnerton-Dyer conjecture ([Tate1966], Conjecture (A))
BSD.weak_birch_swinnerton_dyer_conjecture_rat The weak Birch and Swinnerton-Dyer conjecture over $\mathbb{Q}$, a Clay Millennium Prize Problem
NavierStokes.divergence_of_not_differentiableAt The divergence of a vector field is $0$ at points where fderiv has its junk value
NavierStokes.divergence_zero The divergence of the zero vector field is zero
NavierStokes.divergence_const The divergence of a constant vector field is zero
NavierStokes.divergence_add Divergence is additive at points where both vector fields are differentiable
NavierStokes.divergence_smul Divergence commutes with scalar multiplication at differentiability points
NavierStokes.IsOnePeriodic A function $f : \mathbb{R}^n \to \alpha$ is 1-periodic if it is periodic in each coordinate with period $1$, i.e. $f(x + e_i) = f(x)$ for each unit vector $e_i$
NavierStokes.InitialVelocityCondition Basic conditions on initial velocity field for the Navier-Stokes equations in $n$-dimensional space
NavierStokes.InitialVelocityConditionDecay Initial velocity conditions for the Navier-Stokes problem on all of $\mathbb{R}^n$
NavierStokes.InitialVelocityConditionPeriodic Initial velocity conditions for the periodic Navier-Stokes problem on $\mathbb{R}^n/\mathbb{Z}^n$
NavierStokes.ForceCondition The basic smoothness condition on the external forcing term
NavierStokes.ForceConditionDecay Force conditions for the Navier-Stokes problem on all of $\mathbb{R}^n$
NavierStokes.ForceConditionPeriodic Force conditions for the periodic Navier-Stokes problem on $\mathbb{R}^n/\mathbb{Z}^n$
NavierStokes.NavierStokesExistenceAndSmoothness A solution (v, p) to the Navier-Stokes equations in n-dimensional space with viscosity $\nu$, initial velocity $u_0$, and external force $f$
NavierStokes.NavierStokesExistenceAndSmoothnessRn A solution to the Navier-Stokes equations on all of $\mathbb{R}^n$ with appropriate decay and energy bounds
NavierStokes.NavierStokesExistenceAndSmoothnessPeriodic A solution to the Navier-Stokes equations on the $n$-torus $\mathbb{R}^n/\mathbb{Z}^n$
NavierStokes.navier_stokes_existence_and_smoothness_R3 (A) Existence and smoothness of (unforced) Navier–Stokes solutions on ℝ³
NavierStokes.navier_stokes_existence_and_smoothness_periodic (B) Existence and smoothness of (unforced) Navier–Stokes solutions in ℝ³/ℤ³
PoincareConjecture.ConjectureFor The predicate that the generalized Poincaré conjecture holds in dimension
PoincareConjecture.poincare_conjecture The Millennium Problem, solved by Grigori Perelman in 2003: the Poincaré Conjecture holds
PoincareConjecture.poincare_conjecture.variants.dimension_two The Generalized Poincaré Conjecture holds for surfaces
PoincareConjecture.poincare_conjecture.variants.dimension_ge_five The Generalized Poincaré Conjecture holds for dimensions at least 5
PoincareConjecture.poincare_conjecture.variants.dimension_four The Generalized Poincaré Conjecture holds in dimension 4
PoincareConjecture.SmoothConjectureFor The predicate that the smooth Poincaré conjecture holds in dimension
PoincareConjecture.poincare_conjecture.variants.smooth_for_three A reformulation of the Millennium Problem in terms of smooth 3-folds
PoincareConjecture.poincare_conjecture.variants.smooth_implication The smooth formulation of the Millennium Problem implies the general case
PoincareConjecture.SmoothTrueValues The values at which the smooth version of the conjecture is known to hold
PoincareConjecture.poincare_conjecture.variants.smooth_known_cases The smooth version of the Poincaré conjecture is known to hold in dimensions $1, 2, 3, 5, 6, 12, 56, 61$
PoincareConjecture.poincare_conjecture.variants.smooth_dimension_four The four dimensional case of the smooth version of the conjecture is still open
PoincareConjecture.poincare_conjecture.variants.smooth_other_cases It is conjectured that the only values of $n > 4$ for which the smooth version of the conjecture holds are $n = 5, 6, 12, 56, 61$
ComplexityTheory.P_ne_NP P is not equal to NP
ComplexityTheory.NP_ne_coNP NP ≠ coNP: The conjecture that the complexity classes NP and coNP are not equal
ComplexityTheory.P_eq_BPP P = BPP: The conjecture that the complexity classes P and BPP are equal
ComplexityTheory.coP_eq_P The theorem that the set of complements of languages in P is itself P
ComplexityTheory.P_subset_coNP The theorem that P is a subset of coNP
ComplexityTheory.P_subset_BPP The theorem that P is a subset of BPP
RiemannHypothesis.riemannHypothesis The Riemann Hypothesis
GRH.trivialZeros Let $\chi$ be a Dirichlet character, trivialZeros is the set of trivial zeros of the Dirichlet $L$-function of $\chi$ which is always a set of non-positive integers
GRH.generalized_riemann_hypothesis The Generalized Riemann Hypothesis asserts that all the non-trivial zeros of the Dirichlet
GRH.implies_riemannHypothesis GRH for $\chi = 1$ is RiemannHypothesis
OeisA100434.a The primary defining sequence a, which is the expansion of the generating function $(1+x)(3+x)/(1+6x^2+x^4)$
OeisA100434.c $c(n)$ starts with $(1, -3, -7, 17)$ and satisfies the same recurrence as a
OeisA100434.d $d(n)$ starts with $(2, 4, -10, -24)$ and satisfies the same recurrence as a
OeisA100434.b $b(2n) = -c(2n+1)$, $b(2n+1) = c(2n)$
OeisA100434.e $e(2n) = d(2n)/2$, $e(2n+1) = - d(2n)/2$
OeisA100434.f $f(2n) = f(2n+1) = d(2n+1)/2$
OeisA100434.g $g(2n) = 0, g(2n+1) = c(2n+1)$
OeisA100434.a_0 Value of the sequence a at 0
OeisA100434.a_1 Value of the sequence a at 1
OeisA100434.a_2 Value of the sequence a at 2
OeisA100434.a_3 Value of the sequence a at 3
OeisA100434.a_4 Value of the sequence a at 4
OeisA100434.a_even For all $n \ge 0$, we have $a(2n) = - c(2n+1)$
OeisA100434.a_odd For all $n \ge 0$, we have $a(2n+1) = d(2n+1)$
OeisA100474.triangular The $n$-th triangular number
OeisA100474.a The primary defining sequence a
OeisA100474.a_1 Value of the sequence a at 1
OeisA100474.a_2 Value of the sequence a at 2
OeisA100474.a_3 Value of the sequence a at 3
OeisA100474.a_4 Value of the sequence a at 4
OeisA100474.a_5 Value of the sequence a at 5
OeisA100474.conjecture After $a(2) = 5$, is there another prime?
OeisA100475.reverseDigits Reverses the base 10 digits of a natural number
OeisA100475.a The primary defining sequence a
OeisA100475.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA100475.aStartAt Definition of the generalized sequence starting at x
OeisA100475.IsUltimatelyPeriodic A sequence $f : \mathbb{N} \to \mathbb{N}$ is ultimately periodic if there exist $N, P \in \mathbb{N}$, with $P>0$, such that for all $n \ge N$, $f(n+P) = f(n)$
OeisA100475.aStartAt_zero The totalized recurrence stays at zero when initialized at zero
OeisA100475.conjecture_with_zero If zero is admitted as a starting value, then a start other than $1$ does go into a loop: the sequence starting at zero is constant
OeisA100475.conjecture Starting at a positive value other than $a(0) = 1$, does this sequence ever go into a loop?
OeisA100478.a The primary defining sequence a
OeisA100478.aGeneral A general sequence defined by the Pentanacci $\pi$ recurrence, starting with arbitrary initial values $v: \text{Fin } 5 \to \mathbb{N}$
OeisA100478.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA100478.conjecture Starting with other values of $a(1)$, $a(2)$, $a(3)$, $a(4)$, $a(5)$ what behaviors are possible?
OeisA100800.sumDigits The sum of the decimal digits of a natural number
OeisA100800.f The function $f(n) = n + \text{sum of the digits of } n$
OeisA100800.a a n is the first iteration of $f(n) = n + \text{sum of the digits of } n$ that is a multiple of $n$
OeisA100800.a_of_dvd If n already divides f n, the search stops immediately and a n = f n
OeisA100800.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA100800.conjecture A100800 Conjecture: No term is zero
OeisA101779.Ak A101779: Ak n k is true if for all $i$ from $1$ to $n$, $i \cdot k + (i - 1)$ is prime
OeisA101779.a The primary defining sequence a
OeisA101779.conjecture It is conjectured k always exists
OeisA102371.a The primary defining sequence a
OeisA102371.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA102722.a The primary defining sequence a
OeisA102722.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA102847.a The primary defining sequence a
OeisA102847.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA102847.conjecture Prime for $a(1) = 3$, $a(2) = 11$, $a(4) = 15131$; semiprime for $a(3) = 123 = 3 * 41$, $a(5) = 228947163 = 3 * 76315721$
OeisA103151.a The primary defining sequence a
OeisA103151.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA103151.conjecture Conjecture: all items for $n \ge 4$ are greater than or equal to $1$
OeisA103311.a The sequence $a(n)$ satisfies the linear recurrence relation: $$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$ with initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$
OeisA103425.a The primary defining sequence a
OeisA103425.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA103425.conjecture.variants.nonconstant Is there a non-constant $(a, b, c)$ weighted tribonacci sequence with $a, b, c$ relatively prime which is prime-free?
OeisA103425.conjecture.variants.nonconstant_distinct_coeffs Is there a non-constant $(a, b, c)$ weighted tribonacci sequence with $a, b, c$ relatively prime and pairwise distinct which is prime-free?
OeisA103662.IsValidZerolessPower A helper predicate: $b^n$ is a power with base $>1$ whose decimal representation does not contain the digit 0
OeisA103662.a The primary defining sequence a
OeisA103662.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA103662.conjecture For statistical reasons it is conjectured that the sequence is finite
OeisA103662.conjecture.variants.a_40 $a(40)$, if it exists, is not known
OeisA103885.a The primary defining sequence a
OeisA103885.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA103885.aSubsequenceReal The sequence $b(n) = a(m*n)$ lifted to ℝ
OeisA103885.productIndices The indices $k = 1$ to $2m$, used in the product
OeisA103885.prodFactorPlus The factor $\prod_{k=1}^{2m} (2mn + k)$
OeisA103885.prodFactorMinus The factor $\prod_{k=1}^{2m} (2mn - k)$
OeisA104320.a The primary defining sequence a
OeisA104320.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA104320.conjecture Conjecture from N
OeisA105020.triangularNumber Triangular number $T_c = c(c+1)/2$
OeisA105020.antidiagonalIndex The index $c$ of the antidiagonal containing $n$, i.e., the largest $c$ such that $T_c \le n$
OeisA105020.a The primary defining sequence a
OeisA105020.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA105020.conjecture A "Goldbach Conjecture" for this sequence: when there are
OeisA105033.a The primary defining sequence a
OeisA105033.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA105210.sumRelevantPrimeFactors The sum of distinct prime factors of $n$ that are strictly less than $n$
OeisA105210.a The primary defining sequence a
OeisA105210.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA105210.nextTermFn The transition function for these sequences: $f(n) = n + 1 + S(n)$
OeisA105210.sequenceStartingAt The sequence starting at positive integer $k$, where seq $k$ $n$ is the $n$-th term (1-indexed)
OeisA105210.sequenceSet The set of all terms generated by the sequence starting at $k$
OeisA105210.conjecture_disjoint_starting_values Cormier and Selfridge found 5 starting values for which the sequences appear to not merge
OeisA105210.conjecture This suggests that there may be infinitely many different (non-merging) sequences obtained by choosing different starting values
OeisA105565.a The primary defining sequence a
OeisA105565.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA105565.phi The golden ratio $\phi = (1 + \sqrt{5})/2$
OeisA105565.alphaConst The constant $\alpha = \log(10)/\log(\phi) - 4$
OeisA105565.betaConst The constant $\beta = \log(5)/(2\log(\phi)) - 1$
OeisA105565.s The partial sum $S(n) = \sum_{k=1}^n a(k)$
OeisA105720.a The primary defining sequence a
OeisA105720.a_0 Value of the sequence a at 0
OeisA105720.a_1 Value of the sequence a at 1
OeisA105720.a_2 Value of the sequence a at 2
OeisA105720.a_3 Value of the sequence a at 3
OeisA105720.a_4 Value of the sequence a at 4
OeisA105720.conjecture Terms are squares at only(?) three values of $n = 3, 6, 4072$: corresponding terms are 6^2, 13^2, and 15735^2
OeisA105751.a The primary defining sequence a
OeisA105751.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA105751.prime_divides_some_term Numerical calculation suggests that a similar division holds in this case
OeisA105751.conjecture.variants.moll_p_mod_4_eq_1 Moll's conjecture 5.5 extends to this sequence and takes the form: (ii) for the other primes of type
OeisA105801.a The primary defining sequence a
OeisA105801.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA107247.nonacci Nonacci number (Fibonacci 9-step number)
OeisA107247.a The primary defining sequence a
OeisA107247.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA107247.known_prime_and_semiprimes Primes in this sequence include: $a(8) = 2$
OeisA107247.conjecture Primes in this sequence include: $a(8) = 2$, which is next?
OeisA108.a Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$
OeisA108.catalanReciprocalSum The sum $\sum_{i=j}^k \frac{1}{a(i)}$ of reciprocals of Catalan numbers
OeisA108.IndexCond The index condition on $(j, k)$ from the conjecture: $0 < \min\{2,k\} \le j \le k$
OeisA108.fracPart The fractional part of a rational number
OeisA108081.a The primary defining sequence a
OeisA108081.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA108081.Word Words are finite sequences of integers
OeisA108081.l l w is the word obtained by reversing w and subtracting 1 from every term
OeisA108081.r r w is the word obtained by reversing w and adding 1 to every term
OeisA108081.XWord XWord is the smallest set of words satisfying the inductive properties described in A108081
OeisA108081.xN xN n is the set of words in XWord of length n
OeisA108129.a The primary defining sequence a
OeisA108129.conjecture It is conjectured that the integer $k = 509203$ is the smallest Riesel number, that is, the first $n$ such that $a(n) = -1$ is $254602$
OeisA108211.a The primary defining sequence a
OeisA108211.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA108301.a The primary defining sequence a
OeisA108301.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA108301.primes_in_a $a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes
OeisA108301.conjecture $a(0)$, $a(1)$, $a(5)$, $a(6)$, $a(7)$ and $a(11)$ are primes
OeisA108306.a The primary defining sequence a
OeisA108306.m The matrix for the specific case $a=5, b=2$
OeisA108306.a_is_invert_transform_case The sequence is the INVERT transform of (1, 5, 10, 20, 40, 80, 160, ...) and can be obtained by extracting the upper left terms of matrix powers of [(1,5); (1,2)]
OeisA108306.invertSeqC The c sequence for the general INVERT transform conjecture
OeisA108306.invertSeqD The INVERT transform of the sequence c
OeisA108306.genMatrix The general 2x2 matrix [(1,a); (1,b)]
OeisA108569.A The predicate defining whether $k$ belongs to the sequence
OeisA108569.a The primary defining sequence a
OeisA108569.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA108569.conjecture Conjecture: Except for the first term all terms are even
OeisA108864.perfectDeficiency The perfect deficiency of $n$ (A109883): the remainder after greedily subtracting from $n$ its divisors in increasing order, skipping any divisor larger than the current remainder
OeisA108864.A The condition for a number $n$ to be in the sequence
OeisA108864.a The primary defining sequence a
OeisA108864.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA108864.conjecture Is $1155$ the last odd number in this sequence?
OeisA108866.a The primary defining sequence a
OeisA108866.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA108866.ratExpression The rational number inside the numerator function in the conjecture
OeisA108866.conjecture Conjecture: for $n > 3$, $\textrm{numerator}(-2/n + \sum_{k=1}^{n} \frac{2^k}{k}) == 0 (\textrm{mod} n^2)$ if and only if n is prime
OeisA109074.frac The rational number defined by $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$, whose numerator is A109074
OeisA109074.a The primary defining sequence a
OeisA109074.trinaryLists Lists of length k with entries in {-1, 0, 1}
OeisA109074.isAsmLineAux Checks that all prefix sums of xs starting from acc lie in {0, 1} and the final sum is 1
OeisA109074.isAsmLine A list of integers in {-1, 0, 1} is a valid alternating sign matrix row/column if all of its prefix sums lie in {0, 1} and its total sum is 1
OeisA109074.symAsmRows All vertically symmetric alternating sign matrix rows of length 2 * n + 1
OeisA109074.countVsasmRows Number of ways to complete k remaining vertically symmetric ASM rows given the current column partial sums colSums
OeisA109074.b A005156 (offset 0): the number of $(2n+1) \times (2n+1)$ alternating sign matrices symmetric about the vertical axis (VSASMs)
OeisA109074.conjecture Documented declaration: conjecture
OeisA109227.a The primary defining sequence a
OeisA109227.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA109227.conjecture Conjecture: $a(2)$ and $a(121)$ are primes
OeisA109671.a The primary defining sequence a
OeisA109671.conjecture Does the sequence contain every positive integer (cf. A169741)?
OeisA109845.a $a(1) = 2$; $a(2n)$ = lcm of all previous terms + 1; $a(2n+1)$ = lcm of all previous terms - 1
OeisA109845.conjecture Conjecture: There are infinitely many primes in this sequence
OeisA109905.a $a(n)$ is the greatest prime of the form $k(n-k)+1$, where $k$ can take values from $1$ to $\lfloor n/2 \rfloor$
OeisA109905.conjecture $a(n) = 0$ for $n = 1$, $6$, $30$ and $54$
OeisA109908.a $a(n)$ = greatest prime of the form $k(n-k)-1$, or $0$ if no such prime exists
OeisA109908.conjecture Conjecture: $a(n) > 0$ for $n > 3$
OeisA109909.a $a(n)$ is the number of primes of the form $k(n-k)-1$
OeisA109909.conjecture Conjecture: $a(n) > 0$ for $n > 3$
OeisA110475.a The primary defining sequence a
OeisA110475.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA110475.exceptionalSet The set of exceptional integers
OeisA110475.conjecture It is conjectured that $1,2,3,4,5,6,7,9,11$ are the only positive integers which cannot be represented as the sum of two elements of indices $n$ such that $a(n) = 1$
OeisA110566.a The primary defining sequence a
OeisA110566.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA110566.conjecture It is conjectured that every odd number occurs in this sequence
OeisA110835.a The primary defining sequence a
OeisA110835.a_1 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA110835.conjecture Sierpinski's conjecture (1958) is precisely that $a(n) >= n$ for all $n$
OeisA110854.a The primary defining sequence a
OeisA110854.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA110854.conjecture Do the absolute values cover A004275?
OeisA111114.a a n is the integer part of $\mathrm{prime}(n)/\pi(n)$
OeisA111114.conjecture Conjecture: As $n \rightarrow \infty$, there are infinitely many n's such that $a(n)$ is greater than $a(n+1)$
OeisA111291.countRefactorableNat Helper function: number of refactorable numbers $\le m$
OeisA111291.a a n is the number of refactorable numbers $\le 10^n$
OeisA111291.countRefactorable countRefactorable x is the number of refactorable numbers $\le x$
OeisA111291.conjecture Simon Colton conjectures that the number of refactorable numbers less than $x$ is at least $\frac{x}{2\log x}$
OeisA111291.colton_conjecture Documented declaration: colton_conjecture
OeisA111291.colton_conjecture_le_500 Colton's conjecture holds for $n \le 500$
OeisA112521.a a n is the $n$-th term of the sequence related to NOR bracketings
OeisA112521.T T n k is the array defined as $T(1,1) = 1$, $T(i,j) = 0$ if $i<1$ or $j<1$, $T(n,k) = T(n,k-2) + T(n,k-1) - 2 T(n-1,k-1) + T(n-1,k) + T(n-2,k)$
OeisA112970.a a n is the generalized Stern sequence, defined by the recurrence relations
OeisA112970.conjecture1_value Second part of the conjecture, $a(2^n)=\textrm{A033638}(n)=\lfloor n^2 / 4 \rfloor + 1$
OeisA113010.a a n is the {Number of digits of n} raised to the power of {the sum of the digits of n}
OeisA113010.conjecture $n=1$ and $32$ are two fixed points
OeisA113019.a a n is the (Number of digits of n) raised to the power of (the digital root of n), with appropriate adjustments for $n=0$
OeisA113019.conjecture $n=1$ and $32$ are fixed points
OeisA113213.a a n is the smallest number m such that $2^n - m$ and $2^n + m$ are primes
OeisA113213.conjecture Conjecture: $a(n) = O(n^3)$
OeisA113250.a a n is the sequence defined by the linear recurrence $a(n) = -4 a(n-1) + 64 a(n-3) + 256 a(n-4)$ with initial values $-1, 4, 32, 64$
OeisA113252.a a n is the sequence defined by the linear recurrence $a(n) = -4 a(n-1) + 144 a(n-3) + 1296 a(n-4)$ with initial values $-1, 4, 92, 784$
OeisA113254.a Documented declaration: a
OeisA113255.a a n is the sequence defined by the linear recurrence $a(n) = -4 a(n-1) + 324 a(n-3) + 6561 a(n-4)$ with initial values $-1, 4, 227, 5329$
OeisA113257.a a n is the ascending descending base exponent transform of squares
OeisA113257.conjecture1 The smallest prime in this sequence is $a(2) = 5$
OeisA113257.conjecture2 What is the first square value after 1?
OeisA113258.a The primary defining sequence a
OeisA113258.conjecture Is there a nontrivial power after $a(4) = 5^3$?
OeisA113271.a The primary defining sequence a
OeisA113271.a_5 The next term from the defining formula and the official OEIS b-file
OeisA113271.a_5_not_prime The term a(5) = 135457 = 7 * 37 * 523 is composite
OeisA113271.conjecture1 The first prime terms in this (always odd) sequence are $a(1) = 3$, $a(3) = 41$, and $a(4) = 593$
OeisA113609.IsOeisPrimePower A number $n$ is an "OEIS prime power" (for the context of A113609's definition) if $n=1$ or $n$ is a standard prime power
OeisA113609.a The primary defining sequence a
OeisA113609.conjecture (25,27) is the smallest pair of prime powers (q,q+2) such that both q and q+2 are not primes, conjecture: there are more (but not < 10^6)
OeisA114137.a The primary defining sequence a
OeisA114137.conjecture1 In this powers of 2 sequence, does 1 occur infinitely often?
OeisA114137.conjecture2 Does every odd number occur?
OeisA114216.a The primary defining sequence a
OeisA114216.conjecture Is $a(33900)$ the last term equal to $1$?
OeisA114362.a The primary defining sequence a
OeisA114362.conjecture1 Conjecture: if an integer $n > 1$ is odd, then $\zeta(2n)/\zeta(n)^2$ is irrational
OeisA114362.t t n is used in the second conjecture
OeisA1146.a The primary defining sequence a
OeisA1146.divisibility_fact The forward direction: if $k = a(n)$ for $n \ge 2$, then $k^4 - 1$ divides $2^k - 1$
OeisA1146.conjecture I conjecture that { $a(n)$ ; $n>1$ } are the numbers such that $n^4-1$ divides $2^n-1$, intersection of A247219 and A247165
OeisA114831.a The primary defining sequence a
OeisA115257.a The primary defining sequence a
OeisA115257.conjecture Documented declaration: conjecture
OeisA115366.a The primary defining sequence a
OeisA115366.conjecture Conjecture: $a(n)/A006880(n) \rightarrow 1.77...$ where A006880(n) is the number of primes $\le 10^n$
OeisA11545.a a n is the integer whose decimal digits are the first $n+1$ decimal digits of $\pi$
OeisA11545.conjecture1 Wolfgang Haken (1977) conjectured that no term of this sequence is a perfect square, and estimated the probability that this conjecture is false to be smaller than $10^-9$
OeisA11545.conjecture2 Number of collisions occurring in a system consisting of an infinitely massive, rigid wall at the origin, a ball with mass m stationary at position
OeisA1157.a a n is the sum of squares of divisors of $n$
OeisA1157.conjecture Conjecture: For each k = 2,3,..., all the rational numbers $\frac{\sigma_k(n)}{n^k} = \sum_{d|n} \frac{1}{d^k}$ (n = 1,2,3,...) have pairwise distinct fractional parts
OeisA116150.a a n is the sum of $3^j + (-2)^j$ for j from 1 to n
OeisA116150.conjecture First primes are $a(11) = 264353$ and $a(17) = 193622861$
OeisA117027.a a n is the determinant of a 2x2 matrix of non-overlapping blocks of 4 consecutive primes
OeisA117027.positiveCount The count of positive terms among $a(1)$, ..., $a(N)$
OeisA117027.negativeCount The count of negative terms among $a(1)$, ..., $a(N)$
OeisA117027.ratioSeq The sequence of ratios $P(N)/Neg(N)$ as a sequence of real numbers
OeisA117027.conjecture This suggests the ratio is approaching a limit close to 0.87
OeisA117531.a Number of primes in the $n$-th row of $T(n, k) = k^2 - k + p_n$ for $1 \le k \le n$
OeisA117531.a_0 Value of the sequence a at 0
OeisA117531.a_1 Value of the sequence a at 1
OeisA117531.a_2 Value of the sequence a at 2
OeisA117531.a_3 Value of the sequence a at 3
OeisA117531.a_4 Value of the sequence a at 4
OeisA117531.conjecture Conjecture: $a(n) < n$ for $n > 13$
OeisA117545.a Least $k > 0$ such that $|\Phi_k(n)|$ is prime, or $0$ if no such $k$ exists
OeisA117545.a_1 Value of the sequence a at 1
OeisA117545.a_2 Value of the sequence a at 2
OeisA117545.a_3 Value of the sequence a at 3
OeisA117545.a_4 Value of the sequence a at 4
OeisA117545.conjecture Is $a(n)$ defined for all $n \ge 1$?
OeisA119563.a $a(n) = 2^{2^n} + 2^n - 1$
OeisA119563.a_0 Value of the sequence a at 0
OeisA119563.a_1 Value of the sequence a at 1
OeisA119563.a_2 Value of the sequence a at 2
OeisA119563.a_3 Value of the sequence a at 3
OeisA119563.a_4 Value of the sequence a at 4
OeisA119563.conjecture The first 5 entries are primes
OeisA119591.a Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime, or $0$ if no such $k$ exists
OeisA119591.a_0 Value of the sequence a at 0
OeisA119591.a_1 Value of the sequence a at 1
OeisA119591.a_2 Value of the sequence a at 2
OeisA119591.a_3 Value of the sequence a at 3
OeisA119591.a_4 Value of the sequence a at 4
OeisA119591.a_5 Value of the sequence a at 5
OeisA119591.a_6 Value of the sequence a at 6
OeisA119591.conjecture Is $a(n)$ defined for all $n \ge 2$?
OeisA120424.f Halve even numbers, leave odd numbers unchanged
OeisA120424.a Half-Fibonacci sequence starting with $1, 3$
OeisA120424.a_0 Value of the sequence a at 0
OeisA120424.a_1 Value of the sequence a at 1
OeisA120424.a_2 Value of the sequence a at 2
OeisA120424.a_3 Value of the sequence a at 3
OeisA120424.a_4 Value of the sequence a at 4
OeisA120424.conjecture1 Conjecture (1): The natural density of even terms in the sequence is $1/2$
OeisA120424.conjecture2 Conjecture (2): There are infinitely many consecutive pairs that differ by 1
OeisA1223.a Prime gaps: differences between consecutive primes
OeisA1223.gapSubsequence Helper definition for extracting a finite subsequence (pattern) as a list
OeisA1223.a_0 Value of the sequence a at 0
OeisA1223.a_1 Value of the sequence a at 1
OeisA1223.a_2 Value of the sequence a at 2
OeisA1223.a_3 Value of the sequence a at 3
OeisA129365.a $a(n) = \frac{\prod_{j=1}^n \prod_{k=1}^n \gcd(j,k)}{\prod_{k=1}^n (\lfloor n/k \rfloor!)^k}$
OeisA129365.b Sequence A004125: sum of remainders $n \bmod k$ for $1 \le k \le n$
OeisA129365.a_0 Value of the sequence a at 0
OeisA129365.a_1 Value of the sequence a at 1
OeisA129365.a_2 Value of the sequence a at 2
OeisA129365.a_3 Value of the sequence a at 3
OeisA129365.a_4 Value of the sequence a at 4
OeisA130911.signWeight Parity sign of binary weight: $+1$ if popcount is odd, $-1$ if even
OeisA130911.a $a(n) = \sum_{i=0}^{n-1} \mathrm{signWeight}(p_i)$ where $p_i$ is the $i$-th prime
OeisA130911.a_0 Value of the sequence a at 0
OeisA130911.a_1 Value of the sequence a at 1
OeisA130911.a_2 Value of the sequence a at 2
OeisA130911.a_3 Value of the sequence a at 3
OeisA130911.a_4 Value of the sequence a at 4
OeisA130911.conjecture Shevelev conjectures that $a(n) \ge 0$ for $n > 3$
OeisA135508.x Auxiliary sequence $x(n)$ with $x(1)=1$ and $x(n) = 2 x(n-1) + \operatorname{lcm}(x(n-1),n)$
OeisA135508.a $a(n) = x(n+1)/x(n) - 2$ for $n \ge 1$, and $a(0) = 0$
OeisA135508.a_0 Value of the sequence a at 0
OeisA135508.a_1 Value of the sequence a at 1
OeisA135508.a_2 Value of the sequence a at 2
OeisA135508.a_3 Value of the sequence a at 3
OeisA135508.a_4 Value of the sequence a at 4
OeisA135508.conjecture Conjecture: For prime $p$ such that $p-2$ is not a prime, $a(p-1) = p$
OeisA1359.a The $n$-th lesser twin prime, with $a(0) = 0$
OeisA1359.a_0 Value of the sequence a at 0
OeisA1359.a_1 Value of the sequence a at 1
OeisA1359.a_2 Value of the sequence a at 2
OeisA1359.a_3 Value of the sequence a at 3
OeisA1359.a_4 Value of the sequence a at 4
OeisA141057.a Number of Abelian cubes of length $3n$ over an alphabet of size 3
OeisA141057.a_0 Value of the sequence a at 0
OeisA141057.a_1 Value of the sequence a at 1
OeisA141057.a_2 Value of the sequence a at 2
OeisA141057.a_3 Value of the sequence a at 3
OeisA141057.a_4 Value of the sequence a at 4
OeisA141057.aInt Extension of the sequence $a$ to all integers $n \in \mathbb{Z}$
OeisA141057.aInt_neg_1 Value of the sequence aInt at -1
OeisA141057.aInt_neg_2 Value of the sequence aInt at -2
OeisA141057.conjecture2 Conjecture (Peter Bala, 2022): The supercongruences
OeisA145355.a The sequence $a(n) = \mathrm{round}(\mathrm{round}(\sqrt{n!})/|(\mathrm{round}(\sqrt{n!}))^2 - n!|)$
OeisA145355.a_2 Value of the sequence a at 2
OeisA145355.a_3 Value of the sequence a at 3
OeisA145355.a_4 Value of the sequence a at 4
OeisA145355.a_5 Value of the sequence a at 5
OeisA145355.conjecture This sequence suggests that the distance between a factorial and the closest power is tightly bounded
OeisA153330.collatzStep Single step of the Collatz mapping
OeisA153330.collatzSteps Number of iterations required to turn $n$ into 1 in the Collatz process, or none if $n$ does not terminate
OeisA153330.a The sequence $a(n) = \mathrm{A006577}(n+1) - \mathrm{A006577}(n)$ for $n > 0$, or none if either $n$ or $n+1$ does not terminate
OeisA153330.a_0 Value of the sequence a at 0
OeisA153330.a_1 Value of the sequence a at 1
OeisA153330.a_2 Value of the sequence a at 2
OeisA153330.a_3 Value of the sequence a at 3
OeisA153330.a_4 Value of the sequence a at 4
OeisA153330.indices The set of positive indices $n$ for which $a(n) = v$
OeisA153330.conjecture1 Conjecture 1: More than half of the terms are 0
OeisA153330.conjecture2 Conjecture 2: 1, 6 and 16 appear only once and 3 appears twice in the sequence, i.e., $a(1) = 1$, $a(2) = 6$, $a(4) = a(5) = 3$, and $a(8) = 16$
OeisA153330.conjecture3 Conjecture 3 (Ya-Ping Lu, 2024): Except 1, 3 and 6, the absolute value of all terms can be written as $5x + 8y$ for $x, y \in \mathbb{N}$
OeisA153330.conjecture4 Documented declaration: conjecture4
OeisA157225.a Number of representations of $2n - 1$ as $p + 2^x + 7 \cdot 2^y$ with $p \equiv 5 \pmod 6$
OeisA157225.a_1 Value of the sequence a at 1
OeisA157225.a_2 Value of the sequence a at 2
OeisA157225.a_11 Value of the sequence a at 11
OeisA157225.a_12 Value of the sequence a at 12
OeisA157225.a_13 Value of the sequence a at 13
OeisA157225.a_14 Value of the sequence a at 14
OeisA157237.a Number of representations of $2n - 1$ as $p + 2^x + 11 \cdot 2^y$ with $p \equiv 1 \pmod 6$
OeisA157237.a_1 Value of the sequence a at 1
OeisA157237.a_2 Value of the sequence a at 2
OeisA157237.a_16 Value of the sequence a at 16
OeisA157237.a_17 Value of the sequence a at 17
OeisA157237.a_18 Value of the sequence a at 18
OeisA157237.a_19 Value of the sequence a at 19
OeisA157237.conjecture On Feb
OeisA159829.a $a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime, or none if no such $m$ exists
OeisA159829.a_1 Value of the sequence a at 1
OeisA159829.a_2 Value of the sequence a at 2
OeisA159829.a_3 Value of the sequence a at 3
OeisA159829.a_4 Value of the sequence a at 4
OeisA159829.conjecture2 Conjecture 2: For any $k \ge 3$, there are infinitely many primes of the form $n^k + m^k + 1$ for $n, m \ge 1$
OeisA160324.polygonalNumber $p_k(x) = \frac{(k-2)x(x-1)}{2} + x$ is the $x$-th $k$-gonal number
OeisA160324.pentagonal $p_5(y) = \frac{3y^2 - y}{2}$ is the $y$-th pentagonal number
OeisA160324.hexagonal $p_6(z) = 2z^2 - z$ is the $z$-th hexagonal number
OeisA160324.a Number of representations of $n$ as sum of square, pentagonal, and hexagonal numbers
OeisA160324.a_0 Value of the sequence a at 0
OeisA160324.a_1 Value of the sequence a at 1
OeisA160324.a_2 Value of the sequence a at 2
OeisA160324.a_3 Value of the sequence a at 3
OeisA160324.conjecture1 In April 2009, Zhi-Wei Sun conjectured that $a(n) > 0$ for every $n = 0, 1, 2, 3, \dots$
OeisA160324.conjecture2 For each integer
OeisA160324.conjecture3 The sequence contains every positive integer
OeisA160324.conjecture4 Conjecture (Zhi-Wei Sun, Aug 21 2009): For any integer
OeisA160324.conjecture5 Conjecture (Zhi-Wei Sun, Aug 21 2009): For each integer
OeisA166944.a Defining recurrence for $a(n)$
OeisA166944.a_1 Value of the sequence a at 1
OeisA166944.a_2 Value of the sequence a at 2
OeisA166944.a_3 Value of the sequence a at 3
OeisA166944.a_4 Value of the sequence a at 4
OeisA166944.a_5 Value of the sequence a at 5
OeisA166944.d The difference sequence $d(n) = a(n) - a(n-1)$ for $n \ge 2$
OeisA166944.IsGreaterTwinPrime A prime $p$ is the greater of a twin prime pair if $p$ and $p - 2$ are both prime
OeisA166944.IsDifferenceRecord A value $R$ is a difference record if there exists $n \ge 2$ such that $d(n) = R$ and $R$ is strictly larger than all previous differences $d(k)$ for $2 \le k < n$
OeisA166944.conjecture Conjecture: Every record of differences $a(n)-a(n-1)$ more than 5 is the greater of twin primes (A006512)
OeisA167604.next The least prime occurring among the sums $d + n / d$ for divisors $d$ of $n$
OeisA167604.product Product of the first $n$ terms of Chua's sequence
OeisA167604.a Chua's sequence, extended by $a(0) = 1$
OeisA167604.conjecture Does Chua's sequence contain every prime?
OeisA167918.P $P(i)$ is the $i$-th prime, 1-indexed
OeisA167918.S $S(i) = p_i + p_{i+1}$
OeisA167918.a Smallest index $k > n$ such that $S(n) \mid S(k)$
OeisA167918.a_0 Value of the sequence a at 0
OeisA167918.a_1 Value of the sequence a at 1
OeisA167918.a_2 Value of the sequence a at 2
OeisA167918.a_3 Value of the sequence a at 3
OeisA167918.a_4 Value of the sequence a at 4
OeisA167918.conjecture_infinite Conjecture: the sequence is infinite, that is, for every $n \geq 1$ there is some $k > n$ with $S(n) \mid S(k)$, so that $a(n)$ is defined
OeisA167918.conjecture1 Conjecture: $f(n, k) = 2$ for infinitely many cases, where $k = a(n)$
OeisA167918.conjecture2 Open problem: is the ratio $f(n, k)$ bounded, where $k = a(n)$?
OeisA175386.a $a(n)$ is the denominator of the sum $$\sum_{i=1}^n \frac{1}{i} \binom{2n-i-1}{i-1}$$
OeisA176477.a Rational recurrence sequence $a(n)$
OeisA176477.a_1 Value of the sequence a at 1
OeisA176477.a_2 Value of the sequence a at 2
OeisA17666.a Denominator of sum of reciprocals of divisors of $n$
OeisA17666.conjecture If $a(n)$ is in A005153, then $n$ is in A005153
OeisA179524.a The sequence $a(n) = \sum_{k=0}^n (-4)^k \binom{n}{k}^2 \binom{n-k}{k}^2$
OeisA179524.conjecture1 If $p$ is a prime with $p \equiv 1, 9 \pmod{20}$ and $p = x^2 + 5y^2$ with $x, y$ integers, then $\sum_{k=0}^{p-1} a(k) \equiv 4x^2 - 2p \pmod{p^2}$
OeisA179524.conjecture2 If $p$ is a prime with $p \equiv 3, 7 \pmod{20}$ and $2p = x^2 + 5y^2$ with $x, y$ integers, then $\sum_{k=0}^{p-1} a(k) \equiv 2x^2 - 2p \pmod{p^2}$
OeisA179524.conjecture3 If $p$ is a prime with $p \equiv 11, 13, 17, 19 \pmod{20}$, then $\sum_{k=0}^{p-1} a(k) \equiv 0 \pmod{p^2}$
OeisA179524.conjecture4 $\sum_{k=0}^{n-1}(20k+17)a(k) \equiv 0 \pmod n$ for all $n=1,2,3,\dots$
OeisA179524.conjecture5 $\sum_{k=0}^{p-1}(20k+17)a(k) \equiv p(10(-1/p)+7) \pmod{p^2}$ for any odd prime $p$
OeisA179537.a The sequence $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k$
OeisA179537.a_0 Value of the sequence a at 0
OeisA179537.a_1 Value of the sequence a at 1
OeisA179537.a_2 Value of the sequence a at 2
OeisA179537.a_3 Value of the sequence a at 3
OeisA179537.a_4 Value of the sequence a at 4
OeisA179537.conjecture1 If $p$ is a prime with $(p/7) = 1$ and $p = x^2 + 7y^2$ with $x, y$ integers, then $\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 4x^2 - 2p \pmod{p^2}$
OeisA179537.conjecture2 If $p$ is a prime with $(p/7) = -1$, then $\sum_{k=0}^{p-1} (-1)^k a(k) \equiv 0 \pmod{p^2}$
OeisA179537.conjecture3 $\sum_{k=0}^{n-1} (42k + 37) (-1)^k a(k) \equiv 0 \pmod n$ for all $n \ge 1$
OeisA179537.conjecture4 $\sum_{k=0}^{p-1} (42k + 37) (-1)^k a(k) \equiv p(21(p/7) + 16) \pmod{p^2}$ for any prime $p \ne 7$
OeisA180017.a Difference of sums of digits of $n$ in ternary and in binary
OeisA180017.conjecture "This sequence is positive on average, since 1/log(3) > 1/log(4)
OeisA181546.F The generalized sum $F(n, L) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^L$
OeisA181546.a $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4$
OeisA181546.lucas Lucas numbers $L(0) = 2, L(1) = 1, L(n) = L(n-1) + L(n-2)$
OeisA181546.limitValue Conjectured limit value for $F(n+1, L) / F(n, L)$
OeisA181546.a_0 Value of the sequence a at 0
OeisA181546.a_1 Value of the sequence a at 1
OeisA181546.a_2 Value of the sequence a at 2
OeisA181546.a_3 Value of the sequence a at 3
OeisA181546.a_4 Value of the sequence a at 4
OeisA1818.a The sequence of squares of double factorials: $a(n) = ((2n-1)!!)^2$
OeisA1818.a_0 Value of the sequence a at 0
OeisA1818.a_1 Value of the sequence a at 1
OeisA1818.a_2 Value of the sequence a at 2
OeisA1818.a_3 Value of the sequence a at 3
OeisA1818.a_4 Value of the sequence a at 4
OeisA1818.fEntry Characteristic function $f(j, k)$ for matrix entries in $\mathbb{Z}/p^2\mathbb{Z}$
OeisA1818.conjecture1 Conjecture 1: For any primitive
OeisA1818.conjecture2 Conjecture 2: Let $p$ be an odd prime
OeisA182126.prime $\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1) = 2$)
OeisA182126.a $a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2)$
OeisA182126.countA Count of occurrences of value $v$ among $a(1), \dots, a(x)$
OeisA182126.IsMostFrequent $v_0$ is a most frequent value among $a(1), \dots, a(x)$
OeisA182126.conjecture1 Conjecture: For $x > 10^9$, the most frequent value in $a(n)$, $n=1\dots x$, has form $120k$
OeisA182126.conjecture2 Let $b = \mathrm{prime}(n+2) - \mathrm{prime}(n)$ and $c = \mathrm{prime}(n+2) - \mathrm{prime}(n+1)$
OeisA182126.conjecture3 Are 2, 7, 11, 13, 29 the only primes in this sequence?
OeisA182510.a Defining recurrence for $a(n)$
OeisA182510.conjecture1 Conjecture: the sequence contains 8 zeros
OeisA182510.conjecture2 Conjecture: more positive terms than negative
OeisA185150.a Number of odd primes $p \in (n^2, (n+1)^2)$ with $(n/p) = 1$
OeisA185150.a_0 Value of the sequence a at 0
OeisA185150.a_1 Value of the sequence a at 1
OeisA185150.a_2 Value of the sequence a at 2
OeisA185150.a_3 Value of the sequence a at 3
OeisA185150.a_4 Value of the sequence a at 4
OeisA185150.conjecture Conjecture: $a(n) > 0$ for all $n > 0$
OeisA185895.P The finite polynomial approximation $\prod_{k=1}^n (1 - X^k / k!)$
OeisA185895.a The sequence $a(n) = n! [x^n] \prod_{k=1}^n (1 - x^k / k!)$
OeisA185895.IsTriangular A natural number $n$ is triangular if $n = k(k+1)/2$ for some $k \in \mathbb{N}$
OeisA185895.a_0 Value of the sequence a at 0
OeisA185895.a_1 Value of the sequence a at 1
OeisA185895.a_2 Value of the sequence a at 2
OeisA185895.a_3 Value of the sequence a at 3
OeisA185895.a_4 Value of the sequence a at 4
OeisA185895.c The $n$-th coefficient of the square of the ordinary generating function $A(x)^2$
OeisA185895.conjecture2 The coefficients $c(n)$ of $A(x)^2 = (\sum_{n \ge 0} a(n) x^n)^2$ differ in sign from $c(n-1)$ if and only if $n$ is a triangular number
OeisA194806.setProd The set of all products of elements from a Finset S
OeisA194806.a Size of the smallest subset $S$ of $T = \{1,2,3,\dots,n\}$ such that $S \cdot S$ contains $T$, where $S \cdot S$ is the set of all products of elements of $S$
OeisA211417.a Integral factorial ratio sequence: $$a(n) = \frac{(30n)! n!}{(15n)! (10n)! (6n)!}$$
OeisA211417.divisorProduct The product term in the denominator of the general conjecture
OeisA22030.a Documented declaration: a
OeisA224.a The number of squares modulo $n$
OeisA224.a_0 Value of the sequence a at 0
OeisA224.a_1 Value of the sequence a at 1
OeisA224.a_2 Value of the sequence a at 2
OeisA224.a_3 Value of the sequence a at 3
OeisA224.a_4 Value of the sequence a at 4
OeisA224.conjecture $n^2 \equiv 1 \pmod{a(n)(a(n)-1)}$ if and only if $n$ is an odd prime
OeisA224515.a a: $a(n) = \text{least } k \text{ such that } \sqrt{k^2 \operatorname{XOR} (k+1)^2} = 2n+1$, $a(n) = -1 \text{ if there is no such } k$
OeisA227582.baseSeq The sequence $b_n$ such that $A227582(n) = b_{n-1}$ for $n \ge 1$
OeisA227582.a Expansion of $(2+3x+2x^2+2x^3+3x^4+x^5-x^6)/(1-2x+x^2-x^5+2x^6-x^7)$, 1-indexed
OeisA227582.baseSeq_recurrence The linear recurrence: $$b(n + 7) = 2 b(n + 6) - b(n + 5) + b(n + 2) - 2 b(n + 1) + b(n)$$ holds for baseSeq
OeisA228143.aperyLike The auxiliary sequence used for the Hankel matrix, defined as $$\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$$
OeisA228143.a Determinant of the $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\dots,n$
OeisA228143.ogfAScaled The power series $A(x/3) = \sum_{n=0}^\infty \frac{a(n)}{3^n} x^n$ over ℚ
OeisA228828.a Numbers $n$ such that $n^2 + \pi(n)$ is prime
OeisA228828.conjecture Conjecture: the sequence A228828 is infinite
OeisA231201.A The predicate that n can be written as $x + y$ with $x,y >0$ such that $2^x + y$ is prime
OeisA231201.conjecture The conjecture for sequence A231201: for any $n > 1$, there exist $x, y > 0$ such that $n = x + y$ and $2^x + y$ is prime
OeisA232174.A The predicate that n can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime
OeisA232174.conjecture Zhi-Wei Sun's Conjecture (A232174): Any integer $n > 1$ can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime
OeisA2326.a The multiplicative order of 2 modulo $2n+1$
OeisA2326.a_0 Value of the sequence a at 0
OeisA2326.a_1 Value of the sequence a at 1
OeisA2326.a_2 Value of the sequence a at 2
OeisA2326.a_3 Value of the sequence a at 3
OeisA2326.a_4 Value of the sequence a at 4
OeisA2326.conjecture1 If $p$ is an odd prime then $a((p^3-1)/2) = p \cdot a((p^2-1)/2)$
OeisA2326.conjecture2 A generalization of the previous conjecture: For each $k \ge 2$, if $p$ is an odd prime then $a((p^{k+1}-1)/2) = p \cdot a((p^k-1)/2)$
OeisA237271.a Number of parts in the symmetric representation of $\sigma(n)$
OeisA237271.num2DenseSublists Number of maximal contiguous sublists of divisors of n where each adjacent pair (d_k, d_{k+1}) satisfies d_{k+1} <= 2 * d_k
OeisA237271.A001227 Number of odd divisors of n (A001227)
OeisA237271.A239657 Number of odd divisors m of n such that there is a divisor d of n with d < m < 2*d (A239657)
OeisA237271.conjecture_1 Conjecture 1 / Theorem: "a(n) is the number of odd divisors of n except the 'e' odd divisors described in A005279." - _Omar E
OeisA237271.a_odd_prime_pow Theorem: "a(p^k) = k + 1, where p is an odd prime and k >= 0." - _Hartmut F
OeisA237271.conjecture_3 Conjecture 3: "a(n) is the number of divisors p of n such that p is greater than twice the adjacent previous divisor of n
OeisA237271.observation_carmichael Observation: "a(A002997(n)) >= 3, at least for 1 <= n <= 10000." - _Omar E
OeisA239957.A The predicate that $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer
OeisA239957.conjecture Zhi-Wei Sun's Conjecture (A239957): Every prime $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer
OeisA2407.A A natural number is in A002407 when it is prime and is the difference of two consecutive positive cubes
OeisA2407.conjecture This sequence is believed to be infinite
OeisA2426.a Central trinomial coefficients: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} \binom{2k}{k}$
OeisA2426.a_0 Value of the sequence a at 0
OeisA2426.a_1 Value of the sequence a at 1
OeisA2426.a_2 Value of the sequence a at 2
OeisA2426.a_3 Value of the sequence a at 3
OeisA2426.a_4 Value of the sequence a at 4
OeisA2426.conjecture An integer $n > 3$ is prime if and only if $a(n) \equiv 1 \pmod{n^2}$
OeisA243106.a The sequence $$a(n) = \sum_{k=1}^n (-1)^{\operatorname{isprime}(k)} 10^k$$ where the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime
OeisA24356.a The determinant of the $n \times n$ Hankel matrix of primes
OeisA24356.a_0 Value of the sequence a at 0
OeisA24356.a_1 Value of the sequence a at 1
OeisA24356.a_2 Value of the sequence a at 2
OeisA24356.a_4 Value of the sequence a at 4
OeisA24356.conjecture "I conjecture that $a(4)$ is the only zero
OeisA2454.a Central factorial numbers: $a(n) = 4^n (n!)^2$
OeisA2454.a_0 Value of the sequence a at 0
OeisA2454.a_1 Value of the sequence a at 1
OeisA2454.a_2 Value of the sequence a at 2
OeisA2454.a_3 Value of the sequence a at 3
OeisA2454.a_4 Value of the sequence a at 4
OeisA2454.conjecture Let $\zeta$ be a primitive $(2n+1)$-th root of unity
OeisA248802.a Smallest prime factor of $2^{2^n+2} + 3$
OeisA248802.CoveredByC1 An index k is covered by Conjecture 1 if k = 10m + 2 for some m >= 0, predicting a(k)=67
OeisA248802.CoveredByC2 An index k is covered by Conjecture 2 if k = 36m + 16 for some m >= 0, and m is not 1 mod 5, predicting a(k)=271
OeisA248802.CoveredByC3 An index k is covered by Conjecture 3 if k = 84m + 22 for some m >= 0, and m is not 0 mod 5, predicting a(k)=523
OeisA256012.a a: Number of partitions of $n$ into distinct parts that are not squarefree
OeisA258667.inner_sum The inner sum of the formula used in a: $$\sum_{\max(k-n+5, 0) \le j \le \min(k,4)} \binom{8-j}{j}\binom{2n-k+j-10}{k-j}$$
OeisA258667.a a: A total of $n$ married couples, including a mathematician M and his wife, are to be seated at the $2n$ chairs around a circular table, with no man seated next to his wife
OeisA258667.menageDenomTerm The denominator term $k! (n-1)_k$ represented as a Real number
OeisA258667.asymptoticSumPart The infinite series part of the asymptotic expansion: $\sum_{k \ge 1} \frac{(-1)^k}{k!(n-1)_k}$
OeisA258667.asymptoticTerm The proposed asymptotic expression for a(n)
OeisA260194.a OEIS A260194, shifted so that Lean index zero is the first OEIS term, with recurrence $a(n+3) = a(n+2) + \gcd(a(n+2),a(n))$ and initial values $a(0)=a(1)=a(2)=1$
OeisA260194.conjecture Does every positive integer occur as a difference in this sequence?
OeisA267581.caRule167 The rule function for Rule 167
OeisA267581.caState The state of the Rule 167 elementary cellular automaton at time $t$ and position $x$
OeisA267581.middleColumnBit The sequence of bits forming the middle column of the CA pattern, $C_{t, 0}$
OeisA267581.a Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell
OeisA267581.oeisFloorTerm The floor term in the conjectured recurrence relation for A267581
OeisA271591.tribonacci The Tribonacci numbers $T_n$ (A000073)
OeisA271591.a Second most significant bit of the tribonacci number A000073(n)
OeisA278070.a a: $a(n) = \text{hypergeometric}([n, -n], [], -1)$
OeisA280831.A The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square
OeisA280831.conjecture Documented declaration: conjecture
OeisA281976.A The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers, $z \leq w$, such that both $x$ and $x + 24y$ are squares
OeisA281976.conjecture Zhi-Wei Sun's Conjecture (A281976): Any integer
OeisA282779.a Period of cubes mod $n$
OeisA282779.periodOfPowerMod The length of the minimal positive period of the sequence $k^p \pmod n$
OeisA287616.A The predicate that n can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ for nonnegative integers $x, y, z$
OeisA287616.conjecture Zhi-Wei Sun's Conjecture (A287616): Any nonnegative integer can be written as the sum of a triangular number
OeisA28859.a A028859 (OEIS): $a(n+2) = 2 \cdot a(n+1) + 2 \cdot a(n)$; $a(0) = 1$, $a(1) = 3$
OeisA289411.a a: $\mathrm{a}(n) = \sum_{k=0}^n \mathrm{sign}(\mathrm{A007953}(5k) - \mathrm{A007953}(k))$
OeisA2897.a The sequence $a(n)$ is defined by $a(n) = \binom{2n}{n}^3$
OeisA2897.xyzPowN The finsupp corresponding to the monomial $x^n y^n z^n$
OeisA2897.pPoly The polynomial $pPoly(X, Y, Z) = (1 + X + Y + Z)^{2n} (1 + X + Y - Z)^n (1 + X - Y + Z)^n$
OeisA300997.a Documented declaration: a
OeisA303639.A The predicate that $n$ can be written as $a^2 + b^2 + \binom{2c+1}{c} + \binom{2d+1}{d}$ for nonnegative integers $a, b, c, d$
OeisA303656.A The predicate that n can be written as $a^2 + b^2 + 3^c + 5^d$ for nonnegative integers
OeisA303656.conjecture Zhi-Wei Sun's Conjecture (A303656): Any integer $n > 1$ can be written as the sum of two squares, a power of 3, and a power of 5
OeisA306424.Condition Numbers $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits
OeisA306424.a The $n$-th number $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits
OeisA306477.A The predicate that n can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$
OeisA307865.a $a(n)$ is the number of natural bases $b < 2n+1$ such that $b^n \equiv -1 \pmod{2n+1}$
OeisA307865.IsAbsoluteEulerPseudoprime A natural number $m > 1$ is an absolute Euler pseudoprime if it is composite and for all $b$ coprime to $m$, $b^{(m-1)/2} \equiv \pm 1 \pmod m$
OeisA308734.A The predicate that n can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$
OeisA308734.conjecture Zhi-Wei Sun's Four-Square Conjecture (A308734): Any integer
OeisA309132.a $a(n)$ is the denominator of $F(n)$ = A027641(n-1)/n + A027642(n-1)/n^2
OeisA309132.IsCarmichaelNumber Definition of a Carmichael number $n$: a composite number s.t
OeisA309132.IsComposite Helper definition for "composite number"
OeisA3161.a A binomial coefficient sum: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} (\binom{n}{k} - \binom{n}{k-1})^3$
OeisA3161.b Auxiliary sequence $b(n) = a(2n-1)$
OeisA3161.a_0 Value of the sequence a at 0
OeisA3161.a_1 Value of the sequence a at 1
OeisA3161.a_2 Value of the sequence a at 2
OeisA3161.a_3 Value of the sequence a at 3
OeisA3161.a_4 Value of the sequence a at 4
OeisA3161.conjecture Let $b(n) = a(2n-1)$
OeisA3162.a A binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$
OeisA3162.b Auxiliary sequence $b(n) = a(2n-1)$
OeisA3162.a_0 Value of the sequence a at 0
OeisA3162.a_1 Value of the sequence a at 1
OeisA3162.a_2 Value of the sequence a at 2
OeisA3162.a_3 Value of the sequence a at 3
OeisA3162.a_4 Value of the sequence a at 4
OeisA3162.a_is_integer $a(n)$ is an integer for all $n \ge 0$
OeisA3162.conjecture Let $b(n) = a(2n-1)$
OeisA317940.a005187 A005187: the sum of $\lfloor e / 2^k \rfloor$ over $k \ge 0$
OeisA317940.a046644 A046644: the multiplicative function whose value at each prime power $p^e$ is $2^{\operatorname{A005187}(e)}$
OeisA317940.f The rational sequence whose Dirichlet convolution square is A046644, written using the defining recurrence from the OEIS entry
OeisA317940.a A317940: the numerator of f n
OeisA317940.f_eq The defining recurrence of f, with the recursive calls folded back to f
OeisA317940.f_prime At a prime p every divisor is 1 or p, so the interior sum is empty
OeisA317940.f_four 4 = 2 ^ 2 is the only case here with a nonempty interior sum, contributing f 2 * f 2
OeisA317940.a_1 f 1 = 1 is the base case of the recurrence, so a 1 = 1
OeisA317940.a_2 2 is prime, so the interior sum is empty and f 2 = a046644 2 / 2 = 1
OeisA317940.a_3 3 is prime, so the interior sum is empty and f 3 = a046644 3 / 2 = 1
OeisA317940.a_4 4 is the first index with a nonempty interior sum, giving f 4 = 7 / 2
OeisA317940.a_5 5 is prime, so the interior sum is empty and f 5 = a046644 5 / 2 = 1
OeisA323557.a Coefficients of G.f
OeisA325046.a Coefficients of G.f
OeisA340737.a Numerators of a sequence of fractions converging to $e$
OeisA340737.b Denominators of a sequence of fractions converging to $e$
OeisA341254.rConst The constant $r = (2 + \sqrt{5})/2$
OeisA341254.rSq The constant $r^2$
OeisA341254.a $a(n) = \lfloor r \cdot \lfloor r \cdot n \rfloor \rfloor$, where $r = (2 + \sqrt{5})/2$
OeisA34693.a Smallest number $k$ such that $kn + 1$ is prime
OeisA34693.exists_k Conjecture: for every $n > 1$ there exists a number $k < n$ such that $nk + 1$ is a prime
OeisA34693.exists_k_stronger A stronger conjecture: for every n there exists a number $k < 1 + n^{0.75}$ such that $nk + 1$ is a prime
OeisA34693.exists_k_best_possible The expression $1 + n^{0.74}$ does not work as an upper bound
OeisA34693.a_isBigO Conjecture: $a(n) = O(\log(n)\log(\log(n)))$
OeisA34693.a_unbounded Counter-conjecture to a_isBigO: $a(n) / (\log n \log \log n)$ is unbounded
OeisA34694.a Smallest prime $\equiv 1 \pmod n$
OeisA34694.a_1 Value of the sequence a at 1
OeisA34694.a_2 Value of the sequence a at 2
OeisA34694.a_3 Value of the sequence a at 3
OeisA34694.a_4 Value of the sequence a at 4
OeisA34694.a_5 Value of the sequence a at 5
OeisA34694.conjecture "Conjecture: $a(n) < n^2$ for $n > 1$
OeisA357513.a A357513: $a(n) = \text{numerator of } \sum_{k = 1..n} \frac{1}{k^3} \binom{n}{k}^2 \binom{n+k}{k}^2 \text{ for } n \ge 1 \text{ with } a(0) = 0$
OeisA357513.u Let m be a nonnegative integer and set
OeisA357513.general_supercongruence We conjecture that $u(p-1) == 0 (mod p^4)$ for all primes $p$, with a finite number of exceptions that depend on $m$
OeisA358684.a A358684: $a(n)$ is the minimum integer $k$ such that the smallest prime factor of the $n$-th Fermat number exceeds $2^{2^n - k}$
OeisA358684.a' The "original" definition: $a'(n)$ is the minimum $k$ such that $P_n > 2^{2^n - k}$
OeisA358684.log2_minFac_le The log2 of the smallest prime factor of $F_n$ is at most $2^n$
OeisA358684.a_equiv_a' The minimization definition is equivalent to the closed form
OeisA358684.oeis_358684_conjecture_0 Conjecture: the dyadic valuation of A93179(n) - 1 does not exceed 2^n - a(n)
OeisA3625.A A prime $p$ is in A003625 if $p \equiv 3, 5, \text{ or } 6 \pmod 7$
OeisA3625.a_3 $3$ is in the sequence A003625
OeisA3625.a_5 $5$ is in the sequence A003625
OeisA3625.a_13 $13$ is in the sequence A003625
OeisA3625.a_17 $17$ is in the sequence A003625
OeisA3625.a_19 $19$ is in the sequence A003625
OeisA363102.a051403 Auxiliary sequence A051403, defined as $$\frac{(n+2) \sum_{k=0}^n k!}{2}$$
OeisA363102.a Denominator of the continued fraction $1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2)))))$
OeisA363347.continuedFractionDenominator Helper function for a, which computes the denominator $R_k(n)$ of the continued fraction expression
OeisA363347.a Denominator of the continued fraction
OeisA368692.a The factorial ratio $$a(n) = \frac{(12n + 6)! \cdot (6n + 9)!}{108 \cdot (4n + 2)! \cdot (2n + 3)! \cdot ((6n + 5)!)^2}$$ It is conjectured that $a(n)$ are integers
OeisA37274.decimalDigitCount The number of decimal digits of a natural number, counting zero as one digit
OeisA37274.decimalAppend Append the decimal digits of b to those of a
OeisA37274.primeFactorSplice Concatenate the prime factors of n in nondecreasing order, retaining multiplicity
OeisA37274.ReachesPrime A starting value reaches a prime after finitely many prime-factor splicing steps
OeisA37274.home_prime_conjecture Every integer at least two reaches a home prime
OeisA37274.primeFactorSplice_25 The first step in the trajectory from $25$ is $25\mapsto55$
OeisA37274.primeFactorSplice_55 The second step in the trajectory from $25$ is $55\mapsto511$
OeisA37274.primeFactorSplice_511 The third step in the trajectory from $25$ is $511\mapsto773$
OeisA37274.primeFactorSplice_prime A prime is a fixed point of prime-factor splicing
OeisA37274.reachesPrime_25 The trajectory from $25$ reaches the prime $773$ after three steps
OeisA372761.continuedFractionDenominator Recursive helper computing the continued fraction denominator
OeisA372761.a Denominator of the continued fraction $$ \frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{n+4}}}}}} $$
OeisA38098.a Number of primes strictly less than $n^3$
OeisA38098.a_1 Value of the sequence a at 1
OeisA38098.a_2 Value of the sequence a at 2
OeisA38098.a_3 Value of the sequence a at 3
OeisA38098.a_4 Value of the sequence a at 4
OeisA38098.a_5 Value of the sequence a at 5
OeisA38098.conjecture1 Conjecture (i): for any integer $k > 2$, the sequence $\pi(n^k)/n^k$ ($n = 2, 3, \ldots$) is strictly decreasing, where $\pi(x)$ denotes the number of primes not exceeding $x$
OeisA38098.conjecture2 Conjecture (ii): all the numbers $\pi(n^2)/n^2$ ($n = 1, 2, 3, \ldots$) are pairwise distinct
OeisA38107.a Number of primes strictly less than $n^2$
OeisA38107.a_0 Value of the sequence a at 0
OeisA38107.a_1 Value of the sequence a at 1
OeisA38107.a_2 Value of the sequence a at 2
OeisA38107.a_3 Value of the sequence a at 3
OeisA38107.a_4 Value of the sequence a at 4
OeisA38107.a_5 Value of the sequence a at 5
OeisA38107.conjecture Conjecture: all the numbers $\sum_{i=j}^k \frac{1}{a(i)}$ with $1 < j \le k$ have pairwise distinct fractional parts
OeisA382590.abPair Helper function for a, computing the pair $(a(n), b(n))$ such that: $a(n) = a(n-1)b(n-2) + a(n-2)b(n-1)$ $b(n) = a(n-1)b(n-2) - a(n-2)b(n-1)$
OeisA382590.a The sequence defined by the mutual recurrence relations
OeisA382590.kthPrimeFactor The $k$-th smallest distinct prime factor of an integer $n$ (where $k \ge 1$)
OeisA382590.kthPrimeFactor_periodic Conjecture: For any $k > 1$, if you take the $k$-th prime factor of each term, you get an eventually periodic sequence
OeisA38552.HasClassNumber The class number of the imaginary quadratic field $\mathbb{Q}(\sqrt{-k})$ equals $n$
OeisA38552.IsA038552 $k$ is maximal among squarefree numbers such that $\mathbb{Q}(\sqrt{-k})$ has class number $n$
OeisA38552.classNumberOfDiscriminant The class number of the quadratic field with discriminant $d$
OeisA38552.IsLargestNegFundDiscrForClassNumber $|d|$ is the largest absolute value among negative fundamental discriminants with class number $n$
OeisA38552.starkHeegner_classNumberOne The Stark-Heegner theorem [Sta67] implies that the squarefree $k > 0$ such that $\mathbb{Q}(\sqrt{-k})$ has class number $1$ are exactly $\{1, 2, 3, 7, 11, 19, 43, 67, 163\}$
OeisA38552.hasClassNumber_163_1 $\mathbb{Q}(\sqrt{-163})$ has class number $1$
OeisA38552.isA038552_1_163 $163$ is the largest squarefree $k$ with class number $1$
OeisA38552.mod_24_of_isA038552 All terms of A038552 are congruent to $19 \pmod{24}$
OeisA38552.isA038552_eq_largestNegFundDisc A038552 also gives the largest absolute value of negative fundamental discriminant for each class number
OeisA38552.largestEven_lt_largestOdd_negFundDisc For even class number
OeisA38771.a $a(n)$ is the smallest composite number $c$ such that $\textrm{primorial}(n) + c$ is prime
OeisA38771.a_n_exists $a(n) \ne 0$ for all $n$ (i.e., a suitable composite $c$ always exists)
OeisA38771.conjecture1 Conjecture: $\liminf_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 1 <$ $\limsup_{n \to \infty} \frac{a(n)}{p_{n+1}^2} = 2$
OeisA38771.conjecture2 All the terms in this sequence have exactly two prime factors
OeisA40.a The sequence of prime numbers, with $a(0) = 0$ and $a(n) = p_n$ for $n \ge 1$
OeisA40.a_0 Value of the sequence a at 0
OeisA40.a_1 Value of the sequence a at 1
OeisA40.a_2 Value of the sequence a at 2
OeisA40.a_3 Value of the sequence a at 3
OeisA40.a_4 Value of the sequence a at 4
OeisA40.conjecture Conjecture from Thomas Ordowski (2023): $\log \log a(n+1) - \log \log a(n) < 1/n$ for $n > 0$
OeisA41.a The n-th partition number
OeisA41.noPowerPartitionNumber There are no partition numbers $a(k)$ of the form $x^m$, with $x,m$ integers $>1$
OeisA4290.a Least positive multiple of $n$ using only 0's and 1's in base 10
OeisA4290.a_ten_pow $a(10^k) = 10^k$ for all $k$
OeisA4290.conjecture It is known that $a(10^k - 1) = (10^{9k} - 1) / 9$ for all $k$
OeisA46969.a Denominators of coefficients in Stirling's expansion for $\log(\Gamma(z))$
OeisA46969.conjecture1 Conjecture I: if $n > 2$, then $\frac{a(\text{A005382}(n))}{12}$ is prime, where A005382 is the sequence of primes $p$ such that $2p-1$ is also prime
OeisA48153.a $a(n) = \sum_{k=1}^n (k^2 \bmod n)$
OeisA48153.a_1 Value of the sequence a at 1
OeisA48153.a_2 Value of the sequence a at 2
OeisA48153.a_3 Value of the sequence a at 3
OeisA48153.a_4 Value of the sequence a at 4
OeisA48153.a_5 Value of the sequence a at 5
OeisA49473.a Nearest integer to $n/\sqrt{2}$
OeisA49473.zetaThreeReal $\zeta(3)$ (Apéry's constant)
OeisA49473.s Let $s(n) = \zeta(3) - \sum_{k=1}^{n} 1/k^3$
OeisA49473.A001953 A001953: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)\sqrt{2} \rfloor$ for $k \ge 0$
OeisA49473.A001954 A001954: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)(2 + \sqrt{2}) \rfloor$ for $k \ge 0$
OeisA51293.a Number of nonempty subsets of $\{1, 2, 3, \dots, n\}$ whose elements have an integer average
OeisA5153.A A positive integer $n$ is practical if every $m \le n$ can be represented as a sum of distinct divisors of $n$
OeisA5153.a_1 $1$ is a practical number
OeisA5153.a_2 $2$ is a practical number
OeisA5153.a_4 $4$ is a practical number
OeisA5153.a_6 $6$ is a practical number
OeisA5153.a_8 $8$ is a practical number
OeisA5153.conjecture Conjecture: every odd number, beginning with 3, is the sum of a prime number and a practical number
OeisA51903.a Maximum exponent in the prime factorization of $n$
OeisA51903.conjecture1 Are there composite numbers $n > 4$ such that $n \equiv a(n) \pmod{\phi(n)}$?
OeisA51903.conjecture2 Are there odd numbers $n$ such that $a(n) > 1$ and $n \equiv a(n) \pmod{\lambda(n)}$?
OeisA51903.conjecture3 Are there odd numbers $n$ such that $a(n) > 1$ and $n \equiv a(n) \pmod{\operatorname{ord}_n(2)}$?
OeisA5258.a Apéry numbers: $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$
OeisA5258.aperyPoly The polynomial associated with the $n$-th Apéry number: $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$
OeisA5258.conjecture For each $n = 1, 2, 3, \dots$ the polynomial $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$ is irreducible over the field of rational numbers
OeisA52709.a $a(n) = \sum_{k=0}^{n-1} \frac{1}{k+1} \binom{2k}{k} \binom{k}{n-1-k}$
OeisA52709.isPositiveList A list of natural numbers is composed of positive integers
OeisA52709.coversInitialInterval A list covers an initial interval of positive integers if the set of its elements is $\{1, 2, \dots, \max(l)\}$
OeisA52709.hasNondecreasingPattern3 A list has a non-decreasing subsequence of length 3 (the pattern $x \le y \le z$)
OeisA52709.avoidsPatternXYZ A list avoids the pattern $x \le y \le z$ if it does not have a non-decreasing subsequence of length 3
OeisA52709.sequencesCountedByA052709 The set of sequences of length $n-1$ satisfying the conditions of the conjecture
OeisA53000.a $a(n) = (\text{smallest prime} > n^2) - n^2$
OeisA53000.conjecture Conjecture: $a(n) \le 1 + \phi(n)$ for $n > 0$
OeisA53067.triangular The $n$-th triangular number, $T_n = \frac{n(n+1)}{2}$
OeisA53067.concatenateNats Concatenates two natural numbers $a$ and $b$ base 10
OeisA53067.a $a(n)$ is the concatenation of the next $n$ numbers
OeisA53067.conjecture "The second term is a prime
OeisA53175.a Catalan-Larcombe-French sequence
OeisA53175.hankelMatrix The $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry $a(i+j)$
OeisA53175.conjecture Conjecture: let $P(n)$ be the $(n+1) \times (n+1)$ Hankel-type determinant with $(i,j)$-entry equal to $a(i+j)$ for all $i,j = 0, \ldots, n$
OeisA55487.a Least $m$ such that $\phi(m) = n!$
OeisA55487.isFactorialPrime Factorial primes: $n$ such that $n! + 1$ is prime (A002981)
OeisA55487.p The least prime $p > \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime
OeisA55487.q The complementary prime factor $q = \frac{n!}{p - 1} + 1$
OeisA55487.conjecture Conjecture: unless
OeisA56777.A A composite number $n$ is in the sequence A56777 if it satisfies both $\varphi(n+12) = \varphi(n) + 12$ and $\sigma(n+12) = \sigma(n) + 12$
OeisA56777.ComesFromPrimeQuadruple A number $n$ comes from a prime quadruple $(p, p+2, p+6, p+8)$ if $n = p(p+8)$ for some prime $p$ where $p$, $p+2$, $p+6$, $p+8$ are all prime
OeisA56777.a_65 $65$ is in the sequence A56777
OeisA56777.a_209 $209$ is in the sequence A56777
OeisA56777.a_of_comesFromPrimeQuadruple Numbers coming from prime quadruples are in the sequence A56777
OeisA56777.a_11009 $11009$ is in the sequence A56777
OeisA56777.comesFromPrimeQuadruple_of_a All members of the sequence A56777 come from prime quadruples
OeisA56777.mod_72_of_comesFromPrimeQuadruple Numbers coming from prime quadruples satisfy $n \equiv 65 \pmod{72}$
OeisA56777.mod_100_of_comesFromPrimeQuadruple Numbers coming from prime quadruples satisfy $n \equiv 9 \pmod{100}$, except the first value "65"
OeisA60841.lcmMatrix The $n \times n$ matrix with entry $(i,j)$ equal to $1/\operatorname{lcm}(i+1, j+1)$ over $\mathbb{Q}$
OeisA60841.a Numerator of $1/\det(M)$ where $M$ is the $n \times n$ matrix with $M[i,j] = 1/\operatorname{lcm}(i+1,j+1)$
OeisA60841.integerDetN The exceptional values of $n$ where $1/\det(M)$ is conjectured to be an integer
OeisA60841.conjecture1 "Conjecture: $1/\det(M)$ is an integer only for n: 1 to 34, 36 and 38." - _Robert G
OeisA60841.conjecture2 "All denominators are powers of two (A000079)." - _Robert G
OeisA60957.a Number of different products of any subset of $\{1, 2, \dots, n\}$
OeisA60957.productsOfSubsets The set of products of subsets of $\{1, \dots, n\}$
OeisA60957.conjecture Conjecture: let $p \le n$ be prime
OeisA62567.reverseNat The number whose digits in base 10 are $n$'s digits reversed
OeisA62567.a First multiple of $n$ whose reverse is also divisible by $n$, or 0 if no such multiple exists
OeisA63880.unitaryDivisors The set of unitary divisors of $n$: divisors $d$ such that $\gcd(d, n/d) = 1$
OeisA63880.usigma The sum of unitary divisors of $n$, denoted $\text{usigma}(n)$
OeisA63880.A A number $n$ is in the sequence A063880 if $\sigma(n) = 2 \cdot \text{usigma}(n)$
OeisA63880.IsPrimitiveTerm A term $n$ is primitive if no proper divisor of $n$ is in the sequence
OeisA63880.a_108 $108$ is in the sequence A063880
OeisA63880.a_540 $540$ is in the sequence A063880
OeisA63880.a_756 $756$ is in the sequence A063880
OeisA63880.isPrimitiveTerm_108 $108$ is a primitive term
OeisA63880.mod_216_of_a All members of the sequence satisfy $n \equiv 108 \pmod{216}$
OeisA63880.unitaryDivisors_mul_prime The unitary divisors of m * p, for a prime p not dividing m > 0, are the unitary divisors of m and their multiples by p
OeisA63880.usigma_mul_prime usigma is multiplicative at a prime not dividing the argument
OeisA63880.sigma_mul_prime σ is multiplicative at a prime not dividing the argument
OeisA63880.A_mul_prime Membership in the sequence is preserved by multiplying with a prime not dividing the term
OeisA63880.A_of_mul_prime Conversely, if m * p is in the sequence with p a prime not dividing m > 0, so is m
OeisA63880.A_mul_of_squarefree Membership in the sequence is preserved by multiplying with a coprime squarefree number
OeisA63880.powerful_of_isPrimitiveTerm All primitive terms are powerful numbers: if p divided n exactly once, then n / p would also be in the sequence
OeisA63880.unique_primitive_108 $108$ is the only primitive term
OeisA63880.a_of_primitive_mul_squarefree If $m$ is a primitive term and $s$ is squarefree with $\gcd(m, s) = 1$, then $m \cdot s$ is in the sequence
OeisA63880.exists_primitive_of_a Non-primitive terms have the form $m \cdot s$ where $m$ is primitive and $s$ is squarefree with $\gcd(m, s) = 1$
OeisA64169.H The $n$-th harmonic number $H_n = \sum_{k=1}^n \frac{1}{k}$ as a rational number
OeisA64169.a $a(n) = \text{numerator}(H_n) - \text{denominator}(H_n)$
OeisA64313.area The exact area of a regular $n$-gon with side length 1
OeisA64313.a Integer part of area of a regular polygon with $n$ sides each of length 1
OeisA64313.conjecture "Usually (perhaps always?) $\lfloor n^2 / (4\pi) - \pi / 12 \rfloor$ for a polygon of circumference $n$
OeisA6697.morphism The morphism σ on {a, b} defined by a ↦ aab, b ↦ b, represented on Bool where false = a and true = b
OeisA6697.finiteWord The n-th iterate of the morphism applied to [a]
OeisA6697.infiniteWord The infinite word w is the fixed point of the morphism starting from 'a'
OeisA6697.subwordAt A subword (factor) of length n starting at position i
OeisA6697.subwordsOfLength The set of all distinct subwords of length n in the infinite word
OeisA6697.a The subword complexity function: a(n) is the number of distinct subwords of length n
OeisA6697.conjecture Conjecture (A6697): The generating function for the number of subwords of length
OeisA67599.concatenateNats Concatenates two natural numbers $a$ and $b$ base 10
OeisA67599.a Decimal encoding of the prime factorization of $n$
OeisA67599.conjecture "$a(31) = a(177147) = 311$
OeisA67720.A A number $k$ is in the sequence A067720 if $\varphi(k^2 + 1) = k \cdot \varphi(k + 1)$
OeisA67720.a_1 $1$ is in the sequence A067720
OeisA67720.a_2 $2$ is in the sequence A067720
OeisA67720.a_4 $4$ is in the sequence A067720
OeisA67720.a_6 $6$ is in the sequence A067720
OeisA67720.a_8 $8$ is in the sequence A067720
OeisA67720.a_10 $10$ is in the sequence A067720
OeisA67720.a_of_primes If $k + 1$ and $k^2 + 1$ are both prime, then $k$ is in the sequence
OeisA67720.prime_add_one_of_a For members of the sequence other than $8$, we have $k + 1$ is prime
OeisA67836.primeCandidates The finite set in which the least prime strictly above n is sought
OeisA67836.primeCandidates_nonempty Bertrand's postulate makes the finite prime-candidate set nonempty
OeisA67836.leastPrimeAbove The least prime strictly greater than n, found in the Bertrand interval
OeisA67836.leastPrimeAbove_spec The chosen number is strictly above the bound and is prime
OeisA67836.leastPrimeAbove_le The least prime above n is no larger than any other prime above n
OeisA67836.not_prime_of_lt_leastPrimeAbove No number strictly between n and leastPrimeAbove n is prime
OeisA67836.leastPrimeAbove_eq A convenient criterion for computing the least prime above a concrete bound
OeisA67836.State The recursive state records the latest term and the product of all official terms so far
OeisA67836.state The state after n official terms, with the empty state initialized by term and product one
OeisA67836.a OEIS A067836, with the convenience value a 0 = 1 preceding its official first term
OeisA67836.conjecture OEIS A067836 asks: "Do all primes occur in the sequence?"
OeisA67857.a The sequence $a(n) = n! \sum_{d \mid n} \mu(n/d) H_d$ for $n \ge 1$, and $a(0) = 0$
OeisA67857.a_1 Value of the sequence a at 1
OeisA67857.a_2 Value of the sequence a at 2
OeisA67857.a_3 Value of the sequence a at 3
OeisA67857.a_4 Value of the sequence a at 4
OeisA67857.a_5 Value of the sequence a at 5
OeisA69004.a Number of times $n^2 + s^2$ is prime for positive integers $s < n$
OeisA69004.a_1 Value of the sequence a at 1
OeisA69004.a_2 Value of the sequence a at 2
OeisA69004.a_3 Value of the sequence a at 3
OeisA69004.a_4 Value of the sequence a at 4
OeisA69004.a_5 Value of the sequence a at 5
OeisA69004.conjecture1 Conjecture: $a(n) > 0$ for all $n > 1$
OeisA69004.conjecture2 Stronger conjecture: Let $\pi(n)$ be the prime counting function (A000720)
OeisA69004.conjecture2.variants.upper_bound_false The upper bound of the stronger conjecture on its own
OeisA69004.conjecture2.variants.without_upper_bound The part of the stronger conjecture that survives the counterexample to its upper bound
OeisA69922.a Number of primes $p$ such that $n^n \le p \le n^n + n^2$
OeisA69922.a_1 Value of the sequence a at 1
OeisA69922.a_2 Value of the sequence a at 2
OeisA69922.a_3 Value of the sequence a at 3
OeisA69922.a_4 Value of the sequence a at 4
OeisA69922.a_5 Value of the sequence a at 5
OeisA69922.conjecture Question: for any $n > 0$, is there at least one prime $p$ such that $n^n \le p \le n^n + n^2$?
OeisA69923.a Number of primes $p$ such that $2^n \le p \le 2^n + \mathrm{prime}(n)$
OeisA69923.a_1 Value of the sequence a at 1
OeisA69923.a_2 Value of the sequence a at 2
OeisA69923.a_3 Value of the sequence a at 3
OeisA69923.a_4 Value of the sequence a at 4
OeisA69923.conjecture For any $n > 0$, is there always at least one prime $p$ such that $2^n \le p \le 2^n + \mathrm{prime}(n)$?
OeisA7013.a Catalan-Mersenne numbers: $a(0) = 2$, $a(n+1) = 2^{a(n)} - 1$
OeisA7013.conjecture Catalan-Mersenne conjecture: All terms of the Catalan-Mersenne sequence are prime
OeisA70518.a Value of the $n$-th cyclotomic polynomial at $n$
OeisA70518.a_1 Value of the sequence a at 1
OeisA70518.a_2 Value of the sequence a at 2
OeisA70518.a_3 Value of the sequence a at 3
OeisA70518.a_5 Value of the sequence a at 5
OeisA70518.conjecture $a(28341)$ is divisible by $283411^2$
OeisA70823.numDigitsBase10 Number of digits of a natural number $n$ in base 10 (with 1 digit for 0)
OeisA70823.concatenate Concatenate $x$ followed by $y$ in base 10
OeisA70823.a The sequence $a(1)=0, a(2)=1, a(n+2)=|\text{concat}(a(n+1),a(n))-\text{concat}(a(n),a(n+1))|$
OeisA70823.a_1 Value of the sequence a at 1
OeisA70823.a_2 Value of the sequence a at 2
OeisA70823.a_3 Value of the sequence a at 3
OeisA70823.a_4 Value of the sequence a at 4
OeisA70823.a_5 Value of the sequence a at 5
OeisA71524.a Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^2 + (j+1)^2$ is prime, and $0$ otherwise
OeisA71524.a_1 Value of the sequence a at 1
OeisA71524.a_2 Value of the sequence a at 2
OeisA71524.a_3 Value of the sequence a at 3
OeisA71524.a_4 Value of the sequence a at 4
OeisA71524.a_5 Value of the sequence a at 5
OeisA71524.generalDet Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^{2^m} + (j+1)^{2^m}$ is prime, and $0$ otherwise
OeisA71524.conjecture1 Conjecture: $a(n) = 0$ for no $n > 28$
OeisA71524.conjecture2 Conjecture (Generalization): For every $m \in \mathbb{N}$, the determinant generalDet m n is nonzero for all sufficiently large $n$
OeisA71532.a The sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$
OeisA71532.a_1 Value of the sequence a at 1
OeisA71532.a_2 Value of the sequence a at 2
OeisA71532.a_3 Value of the sequence a at 3
OeisA71532.a_4 Value of the sequence a at 4
OeisA71532.a_5 Value of the sequence a at 5
OeisA71532.conjecture2 For $n$ large enough, does $a(n) > \sqrt{n}$ always hold?
OeisA71532.conjecture3 Conjecture: asymptotically, $a(n) \sim C \log(n)^2$ for some constant $C > 0$
OeisA71532.conjecture3_value Conjecture: the constant $C$ in $a(n) \sim C \log(n)^2$ is approximately $1.4$
OeisA72200.a Smallest $k$ such that $k!$ contains exactly $n$ 6's in base 10, or 0 if no such $k$ exists
OeisA72200.a_1 Value of the sequence a at 1
OeisA72200.a_2 Value of the sequence a at 2
OeisA72200.a_3 Value of the sequence a at 3
OeisA72200.a_4 Value of the sequence a at 4
OeisA72200.a_5 Value of the sequence a at 5
OeisA72200.conjecture It is conjectured that $a(24) = 0$ since no factorial less than $10000$ contained just 24 sixes
OeisA7406.a Wolstenholme numbers: numerator of $\sum_{k=1}^n \frac{1}{k^2}$
OeisA7406.conjecture Conjecture: for $n > 3$, $\gcd(n, a(n-1)) = \text{A089026}(n)$
OeisA7468.a Sum of the next $n$ primes, with $a(0) = 0$
OeisA7468.a_0 Value of the sequence a at 0
OeisA7468.a_1 Value of the sequence a at 1
OeisA7468.a_2 Value of the sequence a at 2
OeisA7468.a_3 Value of the sequence a at 3
OeisA7468.conjecture The only positive integer $n$ such that $a(n)$ is a perfect square is $n=38$
OeisA76141.binaryPattern The binary representation of a natural number $n$, most significant bit first
OeisA76141.a Number of times the binary pattern of $n$ occurs as an infix of the binary pattern of $n^2$
OeisA76141.a_0 Value of the sequence a at 0
OeisA76141.a_1 Value of the sequence a at 1
OeisA76141.a_2 Value of the sequence a at 2
OeisA76141.a_3 Value of the sequence a at 3
OeisA76141.a_4 Value of the sequence a at 4
OeisA76141.a_5 Value of the sequence a at 5
OeisA76495.a Smallest positive integer $x$ such that $\sigma_1(x) \bmod x = n$, or $0$ if no such $x$ exists
OeisA76495.a_1 Value of the sequence a at 1
OeisA76495.a_2 Value of the sequence a at 2
OeisA76495.a_3 Value of the sequence a at 3
OeisA76495.a_4 Value of the sequence a at 4
OeisA76495.a_6 Value of the sequence a at 6
OeisA76495.conjecture At present, the 0 entry for $n = 5$ is only a conjecture
OeisA77408.revBase The number whose base-$b$ digits are the reversal of $n$'s base-$b$ digits
OeisA77408.IsBasePalindrome A natural number $n$ is a base-$b$ palindrome if its base-$b$ digits read the same forwards and backwards
OeisA77408.a Trajectory of 103 under the Reverse and Add!
OeisA77408.a_0 Value of the sequence a at 0
OeisA77408.a_1 Value of the sequence a at 1
OeisA77408.a_2 Value of the sequence a at 2
OeisA77408.a_3 Value of the sequence a at 3
OeisA77408.a_4 Value of the sequence a at 4
OeisA77408.a_5 Value of the sequence a at 5
OeisA77408.conjecture $103$ is conjectured to be the smallest number such that the Reverse and Add!
OeisA78590.a $a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$
OeisA78590.a_1 Value of the sequence a at 1
OeisA78590.a_2 Value of the sequence a at 2
OeisA78590.a_3 Value of the sequence a at 3
OeisA78590.a_4 Value of the sequence a at 4
OeisA78590.a_5 Value of the sequence a at 5
OeisA78680.a Smallest $m > 0$ such that $n \cdot 2^m + 1$ is prime, or 0 if no such $m$ exists
OeisA78680.a_1 Value of the sequence a at 1
OeisA78680.a_2 Value of the sequence a at 2
OeisA78680.a_3 Value of the sequence a at 3
OeisA78680.a_4 Value of the sequence a at 4
OeisA78680.a_5 Value of the sequence a at 5
OeisA78680.conjecture There is a conjecture that the first zero is $n = 65536 = 2^{16}$ (which is equivalent to the statement that $2^{2^k} + 1$ is composite for $k > 4$)
OeisA78729.a Least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, or 0 if no such $k$ exists
OeisA78729.a_1 Value of the sequence a at 1
OeisA78729.a_2 Value of the sequence a at 2
OeisA78729.a_3 Value of the sequence a at 3
OeisA78729.a_5 Value of the sequence a at 5
OeisA78729.a_6 Value of the sequence a at 6
OeisA78729.conjecture $(k+1)(k+2)(k+3)(k+4) + 1 = (k^2 + 5k + 5)^2$, which is never prime
OeisA7918.a Least prime $\ge n$ (version 1 of the "next prime" function)
OeisA7918.a_0 Value of the sequence a at 0
OeisA7918.a_1 Value of the sequence a at 1
OeisA7918.a_2 Value of the sequence a at 2
OeisA7918.a_3 Value of the sequence a at 3
OeisA7918.isApOfNPrimes The initial term $p_0$ and common difference $d$ form an arithmetic progression of length $n$ consisting entirely of prime numbers with $d > 0$
OeisA7918.conjecture1 According to the "k-tuple" conjecture
OeisA7918.conjecture2 If $n > 1$, then $a(n) < n^{n^{1/n}}$
OeisA79727.a $a(n) = \sum_{k=0}^n \binom{2k}{k}^3$
OeisA79727.a_0 Value of the sequence a at 0
OeisA79727.a_1 Value of the sequence a at 1
OeisA79727.a_2 Value of the sequence a at 2
OeisA79727.a_3 Value of the sequence a at 3
OeisA79727.a_4 Value of the sequence a at 4
OeisA79727.conjecture1 Conjecture 1 (Peter Bala, 2024): If prime $p$ is in A003625 then $a(p^2) \equiv 8 + p^2 \pmod{p^3}$
OeisA79727.conjecture2 Conjecture 2 (Peter Bala, 2024): If prime $p$ is in A003625 then $a(p(p-1)) \equiv p^2 \pmod{p^3}$
OeisA79727.conjecture3 Conjecture 3 (Peter Bala, 2024): If prime $p$ is in A003625 then $a((p^2-1)/2) \equiv p^2 \pmod{p^4}$
OeisA79727.conjecture4 Conjecture 4 (Peter Bala, 2024): If $n$ is a product of distinct primes from A003625 then $a((n-1)/2)$ is divisible by $n^2$
OeisA80101.a Number of prime powers strictly between the $n$-th prime and the $(n+1)$-th prime
OeisA80101.a_1 Value of the sequence a at 1
OeisA80101.a_2 Value of the sequence a at 2
OeisA80101.a_3 Value of the sequence a at 3
OeisA80101.a_4 Value of the sequence a at 4
OeisA80101.conjecture It is conjectured that $a(n) \le 2$ for all $n$
OeisA80170.A The gcd of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$
OeisA80170.B Let $P$ be the largest prime power dividing k
OeisA80326.term $k^{\mu(k)}$ as a rational number
OeisA80326.a Denominator of $\sum_{k=1}^n k^{\mu(k)}$
OeisA80326.a_1 Value of the sequence a at 1
OeisA80326.a_2 Value of the sequence a at 2
OeisA80326.a_3 Value of the sequence a at 3
OeisA80326.a_4 Value of the sequence a at 4
OeisA80326.a_5 Value of the sequence a at 5
OeisA80326.conjecture Conjecture: $a(n) = \text{primorial}(n)$ for infinitely many $n$
OeisA81091.isPrimeBitsSet Primes with $m$ one bits in their binary representation
OeisA81091.A Primes in A81091 have exactly $3$ set bits in binary representation ($2^n + 2^i + 1$)
OeisA81091.conjectureA81091 Conjecture (A81091): There are infinite primes of the form $2^n + 2^i + 1$, with $0 < i < n$
OeisA83753.IsDecimalPalindrome A natural number $m$ is a decimal palindrome if its base-$10$ digits read the same forwards and backwards
OeisA83753.a Smallest positive palindrome with exactly $n$ divisors, or $0$ if no such number exists
OeisA83753.a_1 Value of the sequence a at 1
OeisA83753.a_2 Value of the sequence a at 2
OeisA83753.a_3 Value of the sequence a at 3
OeisA83753.a_4 Value of the sequence a at 4
OeisA83753.conjecture There are no palindromic numbers greater than 1 which are the fifth or higher power of a natural number
OeisA84046.a Smallest prime $p$ such that $p + n$ is an $n$-th power, or $0$ if no such prime exists
OeisA84046.a_0 Value of the sequence a at 0
OeisA84046.a_1 Value of the sequence a at 1
OeisA84046.a_2 Value of the sequence a at 2
OeisA84046.a_3 Value of the sequence a at 3
OeisA86766.a Sequence $a(n)$ is the smallest $r > 0$ such that the concatenation of $n$, $r$ times with itself, multiplied by $10$ plus $1$, is prime, or $0$ if no such prime exists
OeisA86766.a_0 Value of the sequence a at 0
OeisA86766.a_1 Value of the sequence a at 1
OeisA86766.a_2 Value of the sequence a at 2
OeisA86766.a_3 Value of the sequence a at 3
OeisA86766.a_4 Value of the sequence a at 4
OeisA86766.conjecture1 What is the smallest integer $m > 1$ such that $a(10^m)$ is nonzero?
OeisA86766.conjecture2 Conjecture: If $n$ is not of the form $10^m$ then $a(n)$ is nonzero
OeisA86766.conjecture3 What is the smallest odd prime $p$ such that $(10^{p^2}-1)/(10^p-1)$ is a prime number (and $a(10^{p-1})$ could be nonzero)?
OeisA87207.a Binary representation of the prime factors of $n$, represented in decimal: $a(n) = \sum_{p \in \mathrm{support}(\mathrm{factorization}(n))} 2^{\pi(p) - 1}$
OeisA87207.a_0 Value of the sequence a at 0
OeisA87207.a_1 Value of the sequence a at 1
OeisA87207.a_2 Value of the sequence a at 2
OeisA87207.a_3 Value of the sequence a at 3
OeisA87207.a_4 Value of the sequence a at 4
OeisA87207.conjecture Starting at any $n$ and iterating the map $n \mapsto a(n)$, we will always reach $0$
OeisA87455.a The primary defining sequence a, satisfying $a(n) = 2 a(n-1) - 3 a(n-2)$ for $n \ge 2$, with initial values $a(0)=1$ and $a(1)=1$
OeisA87455.significand The decimal significand of an integer x: the unique real number $S(x) \in [1, 10)$ such that $|x| = S(x) \cdot 10^k$ for some integer $k$, with the convention $S(0) = 0$
OeisA87455.SatisfiesBenford Documented declaration: SatisfiesBenford
OeisA87455.a_0 Value of the sequence a at 0
OeisA87455.a_1 Value of the sequence a at 1
OeisA87455.a_2 Value of the sequence a at 2
OeisA87455.a_3 Value of the sequence a at 3
OeisA87455.a_4 Value of the sequence a at 4
OeisA87455.conjecture It is an open question whether or not this sequence satisfies Benford's law [Berger-Hill, 2017; Arno Berger, email, Jan 06 2017]
OeisA87571.getAllDigitsMsf Concatenate a list of numbers into a list of decimal digits in most-significant-first order
OeisA87571.ofDigitsMsf Convert a list of decimal digits (most significant first) to a natural number
OeisA87571.concatNum Concatenation of numbers $n, n-1, \dots, n-k$
OeisA87571.a Smallest prime of the form $n, n-1, \dots, n-k$ for $k < n$, or $0$ if none exists
OeisA87571.a_0 Value of the sequence a at 0
OeisA87571.a_1 Value of the sequence a at 1
OeisA87571.a_2 Value of the sequence a at 2
OeisA87571.a_3 Value of the sequence a at 3
OeisA87571.a_4 Value of the sequence a at 4
OeisA87571.conjecture Conjecture: There are infinitely many composite numbers $n$ such that $a(n)$ is nonzero
OeisA87719.countExceeding Count of numbers k in {1, ..., m} where k > (minFac k)^n
OeisA87719.countNotExceeding Count of numbers k in {1, ..., m} where k ≤ (minFac k)^n
OeisA87719.a_exists There exists m such that countExceeding n m > countNotExceeding n m
OeisA87719.a The sequence a(n): least m such that countExceeding n m > countNotExceeding n m
OeisA87719.a_1 a(1) = 15
OeisA87719.a_2 a(2) = 27
OeisA87719.a_3 a(3) = 57
OeisA89026.a $a(n) = n$ if $n$ is a prime, otherwise $a(n) = 1$
OeisA89026.a_0 Term theorems verifying the first few values of the sequence against the official OEIS b-file
OeisA91591.a Number of pairs of twin primes $(p, p+2)$ strictly between $n^2$ and $(n+1)^2$
OeisA91591.a_0 Value of the sequence a at 0
OeisA91591.a_1 Value of the sequence a at 1
OeisA91591.a_2 Value of the sequence a at 2
OeisA91591.a_3 Value of the sequence a at 3
OeisA91591.a_4 Value of the sequence a at 4
OeisA91591.conjecture It is conjectured that $a(n)>0$ for all $n>122$
OeisA91669.a $a(n) = \frac{2^{n-1}}{n!} \prod_{k=1}^{n-1} (2^k-1)$
OeisA92243.primeGap The $n$-th prime gap $g(n) = p_n - p_{n-1}$ for $n \ge 1$, where $p_i$ is the $i$-th prime (0-indexed)
OeisA92243.a Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases
OeisA92243.a_0 Value of the sequence a at 0
OeisA92243.a_1 Value of the sequence a at 1
OeisA92243.a_2 Value of the sequence a at 2
OeisA92243.a_3 Value of the sequence a at 3
OeisA92243.conjecture1 Is the score $a(n) > 0$ for some $n > 250000$?
OeisA92243.conjecture2 Is the score $a(n)$ bounded from below?
OeisA92243.conjecture3 Is the score $a(n)$ bounded from above?
OeisA92243.conjecture4 Is the score $a(n) > 0$ for infinitely many values of $n$?
OeisA92243.conjecture5 Is the score $a(n) < 0$ for infinitely many values of $n$?
OeisA93456.a Product of all composite numbers in the interval $[\binom{n}{2} + 1, \binom{n+1}{2}]$
OeisA93456.a_0 Value of the sequence a at 0
OeisA93456.a_1 Value of the sequence a at 1
OeisA93456.a_2 Value of the sequence a at 2
OeisA93456.a_3 Value of the sequence a at 3
OeisA93456.a_4 Value of the sequence a at 4
OeisA93818.a $a(n) = \gcd(\mathrm{numerator}(H_n), n!)$, where $H_n$ is the $n$-th harmonic number
OeisA93818.a_0 Value of the sequence a at 0
OeisA93818.a_1 Value of the sequence a at 1
OeisA93818.a_2 Value of the sequence a at 2
OeisA93818.a_3 Value of the sequence a at 3
OeisA93818.a_4 Value of the sequence a at 4
OeisA93818.conjecture Conjecture: Every odd prime occurs as a term in the sequence
OeisA945.b b n is the product of the first n terms of the Euclid-Mullin sequence
OeisA945.a The Euclid-Mullin sequence, extended by a 0 = 1
OeisA945.every_prime_occurs "Does the sequence
OeisA96535.a Recurrence $a(n) = (a(n-1) + a(n-2)) \pmod n$
OeisA96535.a_0 Value of the sequence a at 0
OeisA96535.a_1 Value of the sequence a at 1
OeisA96535.a_2 Value of the sequence a at 2
OeisA96535.a_3 Value of the sequence a at 3
OeisA96535.a_4 Value of the sequence a at 4
OeisA96535.conjecture All numbers appear infinitely often, i.e., for every number
OpenQuantumProblem13.UMat A unitary matrix representing an orthonormal basis of $\mathbb{C}^d$ via its columns
OpenQuantumProblem13.relativeUnitary The relative unitary between two bases
OpenQuantumProblem13.IsUnbiased Two unitary matrices represent mutually unbiased bases if every entry of the relative unitary has squared norm $1 / d$
OpenQuantumProblem13.IsUnbiased.symm Mutual unbiasedness is symmetric
OpenQuantumProblem13.IsMUBFamily A family of unitary matrices is a family of mutually unbiased bases if every two distinct members are unbiased
OpenQuantumProblem13.HasMUBs There exist $k$ mutually unbiased bases in $\mathbb{C}^d$
OpenQuantumProblem13.HasCompleteMUBs There exists a complete set of $d + 1$ mutually unbiased bases in $\mathbb{C}^d$
OpenQuantumProblem13.IsMaxMUBCount $k$ is the maximal size of a family of mutually unbiased bases in dimension $d$
OpenQuantumProblem13.hasMUBs_zero Every dimension admits the empty family of mutually unbiased bases
OpenQuantumProblem13.hasMUBs_one Every dimension admits a family of one mutually unbiased basis
OpenQuantumProblem13.Qubit.ω A convenient phase with squared norm $1/2$
OpenQuantumProblem13.Qubit.phaseMatrix The raw phase-parametrized Hadamard matrix
OpenQuantumProblem13.Qubit.omega_norm_sq The squared norm of ω is $1/2$
OpenQuantumProblem13.Qubit.conj_omega_mul_omega The product $\overline{\omega}\,\omega$ is $1/2$
OpenQuantumProblem13.Qubit.star_smul_mul_smul Taking the star of a scalar multiple on the left and multiplying by another scalar multiple collects the scalar factor as $\overline{a} a$
OpenQuantumProblem13.Qubit.star_phaseMatrix_mul_phaseMatrix The relative product of two phase matrices has the expected $2 \times 2$ form
OpenQuantumProblem13.Qubit.star_phaseMatrix_mul_self_of_unit_phase If $\zeta$ has unit modulus, then the phase matrix is orthogonal up to the scalar factor $2$
OpenQuantumProblem13.Qubit.scaled_phaseMatrix_mem_unitary Scaling a phase matrix by $\omega$ produces a unitary matrix whenever the phase has unit modulus
OpenQuantumProblem13.Qubit.star_phaseBasis_mul_phaseBasis The relative product of two scaled phase matrices is obtained by scaling the corresponding relative product of phase matrices
OpenQuantumProblem13.Qubit.phaseU The bundled qubit basis associated to a unit-modulus phase $\zeta$
OpenQuantumProblem13.Qubit.phase_norm_sq_eq_one A complex number with $\overline{\zeta}\,\zeta = 1$ has squared norm $1$
OpenQuantumProblem13.Qubit.omega_mul_phase_norm_sq Multiplying $\omega$ by a unit-modulus phase preserves the squared norm $1/2$
OpenQuantumProblem13.Qubit.ZU The standard qubit basis
OpenQuantumProblem13.Qubit.XU The qubit $X$ basis as a bundled unitary matrix
OpenQuantumProblem13.Qubit.YU The qubit $Y$ basis as a bundled unitary matrix
OpenQuantumProblem13.Qubit.isUnbiased_Z_phaseU The standard basis is mutually unbiased with any phase basis of unit-modulus phase
OpenQuantumProblem13.Qubit.relative_phaseU_phaseU_of_mul_eq_I If $\overline{\zeta}\,\eta = i$, then the relative unitary between the corresponding phase bases is the qubit mutually unbiased overlap matrix
OpenQuantumProblem13.Qubit.isUnbiased_phaseU_phaseU_of_mul_eq_I If $\overline{\zeta}\,\eta = i$, then the corresponding phase bases are mutually unbiased
OpenQuantumProblem13.Qubit.qubitFamily The three standard qubit mutually unbiased bases: $Z$, $X$, and $Y$
OpenQuantumProblem13.Qubit.qubitFamily_isMUB The standard qubit family is a family of mutually unbiased bases
OpenQuantumProblem13.Qubit.qubit_hasThreeMUBs There exist three mutually unbiased bases in dimension $2$
OpenQuantumProblem13.Qubit.u0 The first entry of the first column of a qubit unitary basis matrix
OpenQuantumProblem13.Qubit.u1 The second entry of the first column of a qubit unitary basis matrix
OpenQuantumProblem13.Qubit.BlochVec The real Bloch-vector space for qubits
OpenQuantumProblem13.Qubit.bloch The Bloch vector associated to the first column of a qubit basis matrix
OpenQuantumProblem13.Qubit.relativeUnitary_apply_zero_zero The $(0,0)$ entry of the relative unitary is the overlap of the first columns
OpenQuantumProblem13.Qubit.firstCol_normSq The first column of a unitary matrix has squared norm $1$
OpenQuantumProblem13.Qubit.re_mul_conj The real part of $z \overline{w}$ is the Euclidean dot product of the coordinate pairs of z and w
OpenQuantumProblem13.Qubit.bloch_inner_eq_two_normSq_sub_one The Bloch inner product is determined by the $(0,0)$ entry of the relative unitary
OpenQuantumProblem13.Qubit.relativeUnitary_self The relative unitary of a basis with itself is the identity matrix
OpenQuantumProblem13.Qubit.bloch_inner_self Every qubit Bloch vector has squared Euclidean norm $1$
OpenQuantumProblem13.Qubit.bloch_ne_zero A qubit Bloch vector is never the zero vector
OpenQuantumProblem13.Qubit.bloch_inner_eq_zero_of_isUnbiased Mutually unbiased qubit bases have orthogonal Bloch vectors
OpenQuantumProblem13.Qubit.qubit_upper_bound No family of mutually unbiased bases in dimension $2$ has size greater than $3$
OpenQuantumProblem13.Qubit.qubit_maximal The maximum number of mutually unbiased bases in dimension $2$ is $3$
OpenQuantumProblem13.mutuallyUnbiasedBases_dim2 In dimension $2$, the maximum number of mutually unbiased orthonormal bases is $3$
OpenQuantumProblem13.mutuallyUnbiasedBases_dim6 Special case in dimension $6$: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^6$
OpenQuantumProblem13.mutuallyUnbiasedBases_dim10 Special case in dimension $10$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{10}$
OpenQuantumProblem13.mutuallyUnbiasedBases_dim12 Special case in dimension $12$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{12}$
OpenQuantumProblem13.mutuallyUnbiasedBases_dim14 Special case in dimension $14$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{14}$
OpenQuantumProblem13.mutuallyUnbiasedBases_dim15 Special case in dimension $15$ (not a prime power): determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^{15}$
OpenQuantumProblem13.mutuallyUnbiasedBases Open Quantum Problem 13: determine the maximal number of mutually unbiased orthonormal bases in $\mathbb{C}^d$ for $d \ge 2$
OpenQuantumProblem23.StateVector A state vector in the $d$-dimensional complex Hilbert space $\mathbb{C}^d$
OpenQuantumProblem23.mkStateVector Build a state vector from its coordinates in the computational basis
OpenQuantumProblem23.IsNormalized A state vector is normalized if it has $L^2$ norm $1$
OpenQuantumProblem23.overlapSq The squared magnitude of the overlap between two state vectors
OpenQuantumProblem23.HasConstantOverlapSq A family has constant pairwise squared overlap $c$ if every two distinct members have squared overlap $c$
OpenQuantumProblem23.sicOverlapSq The squared overlap value of a SIC family in dimension $d$
OpenQuantumProblem23.IsSICFamily A SIC family in dimension $d$ consists of $d^2$ normalized vectors in $\mathbb{C}^d$ with pairwise squared overlap $(d + 1)^{-1}$
OpenQuantumProblem23.HasSICPOVM There exists a SIC-POVM in dimension $d$
OpenQuantumProblem23.hasConstantOverlapSq_singleton Any singleton family has constant pairwise squared overlap, vacuously
OpenQuantumProblem23.sicOverlapSq_one The SIC overlap value in dimension $1$ is $1/2$
OpenQuantumProblem23.sicOverlapSq_pos The SIC overlap value is positive in every dimension
OpenQuantumProblem23.isSICFamily_singleton_iff In dimension $1$, a singleton family is SIC exactly when its vector is normalized
OpenQuantumProblem23.hasSICPOVM_zero The empty family witnesses the degenerate dimension-$0$ case
OpenQuantumProblem23.isSICFamily_one_of_normalized Any normalized state in dimension $1$ yields a SIC family
OpenQuantumProblem23.hasSICPOVM_one Dimension $1$ admits a SIC-POVM
OpenQuantumProblem23.ω The standard algebraic primitive cube root of unity
OpenQuantumProblem23.tetraA The first real amplitude used in the tetrahedral qubit SIC
OpenQuantumProblem23.tetraB The second real amplitude used in the tetrahedral qubit SIC
OpenQuantumProblem23.hesseS The common scale used in the Hesse qutrit SIC
OpenQuantumProblem23.vec2 A convenient constructor for qubit state vectors
OpenQuantumProblem23.vec3 A convenient constructor for qutrit state vectors
OpenQuantumProblem23.qubitSICFamily The tetrahedral qubit SIC family
OpenQuantumProblem23.hesseFamily The Hesse qutrit SIC family
OpenQuantumProblem23.bb84Family The BB84 family of four qubit states
OpenQuantumProblem23.sicOverlapSq_two The SIC overlap value in dimension $2$ is $1/3$
OpenQuantumProblem23.sicOverlapSq_three The SIC overlap value in dimension $3$ is $1/4$
OpenQuantumProblem23.omega_norm The unit complex number ω (primitive cube root) has norm 1
OpenQuantumProblem23.qubitSICFamily_normalized Every vector in the tetrahedral qubit SIC family is normalized
OpenQuantumProblem23.qubitSICFamily_pairwise The tetrahedral qubit SIC family has the correct constant pairwise overlap
OpenQuantumProblem23.hasSICPOVM_two Dimension $2$ admits a SIC-POVM, witnessed by the tetrahedral qubit SIC
OpenQuantumProblem23.hesseFamily_normalized Every vector in the Hesse qutrit SIC family is normalized
OpenQuantumProblem23.hesseFamily_pairwise The Hesse qutrit SIC family has the correct constant pairwise overlap
OpenQuantumProblem23.hasSICPOVM_three Dimension $3$ admits a SIC-POVM, witnessed by the Hesse qutrit SIC
OpenQuantumProblem23.bb84Family_normalized Every vector in the BB84 family is normalized
OpenQuantumProblem23.bb84Family_not_isSICFamily The BB84 family has the right cardinality for a qubit SIC but fails the constant-overlap condition
OpenQuantumProblem23.hasSICPOVM_56 Benchmark open subproblem: existence of a SIC-POVM in dimension $56$
OpenQuantumProblem23.hasSICPOVM_58 Benchmark open subproblem: existence of a SIC-POVM in dimension $58$
OpenQuantumProblem23.hasSICPOVM_59 Benchmark open subproblem: existence of a SIC-POVM in dimension $59$
OpenQuantumProblem23.hasSICPOVM_60 Benchmark open subproblem: existence of a SIC-POVM in dimension $60$
OpenQuantumProblem23.hasSICPOVM_64 Benchmark open subproblem: existence of a SIC-POVM in dimension $64$
OpenQuantumProblem23.hasSICPOVM_68 Benchmark open subproblem: existence of a SIC-POVM in dimension $68$
OpenQuantumProblem23.hasSICPOVM_69 Benchmark open subproblem: existence of a SIC-POVM in dimension $69$
OpenQuantumProblem23.hasSICPOVM_70 Benchmark open subproblem: existence of a SIC-POVM in dimension $70$
OpenQuantumProblem23.hasSICPOVM_71 Benchmark open subproblem: existence of a SIC-POVM in dimension $71$
OpenQuantumProblem23.hasSICPOVM_72 Benchmark open subproblem: existence of a SIC-POVM in dimension $72$
OpenQuantumProblem23.hasSICPOVM_75 Benchmark open subproblem: existence of a SIC-POVM in dimension $75$
OpenQuantumProblem23.sicPOVMs Do SIC-POVMs exist in every finite dimension?
OpenQuantumProblem35.Config A computational-basis configuration of $n$ parties with local dimension $d$
OpenQuantumProblem35.StateVector A state vector in the computational basis, viewed as a finite-dimensional Hilbert space
OpenQuantumProblem35.mkStateVector Build a state vector from its computational-basis amplitudes
OpenQuantumProblem35.mkStateVector_apply A state built from amplitudes has those amplitudes as its coordinates
OpenQuantumProblem35.IsNormalized A state vector is normalized if it has $L^2$ norm $1$
OpenQuantumProblem35.isNormalized_iff_norm_sq_eq_one A state is normalized iff its squared $L^2$ norm is $1$
OpenQuantumProblem35.permuteConfig Permute the parties of a configuration
OpenQuantumProblem35.permuteConfig_refl The identity permutation leaves a configuration unchanged
OpenQuantumProblem35.permuteState Permute the parties of a state vector
OpenQuantumProblem35.permuteState_apply Evaluating a permuted state vector reads the amplitude at the permuted configuration
OpenQuantumProblem35.permuteState_refl The identity permutation leaves a state vector unchanged
OpenQuantumProblem35.combineFirst Merge a configuration on the first $m$ parties and a configuration on the remaining $n-m$ parties into a configuration on all $n$ parties
OpenQuantumProblem35.leftIndex The embedding of the first $m$ indices into $\mathrm{Fin}\, n$
OpenQuantumProblem35.rightIndex The embedding of the last $n-m$ indices into $\mathrm{Fin}\, n$
OpenQuantumProblem35.combineFirst_leftIndex Combining and then restricting to the left block recovers the left input
OpenQuantumProblem35.combineFirst_rightIndex Combining and then restricting to the right block recovers the right input
OpenQuantumProblem35.reducedDensityFirst The reduced density matrix obtained by tracing out the last $n-m$ parties
OpenQuantumProblem35.maximallyMixed The maximally mixed state on $m$ parties
OpenQuantumProblem35.HasMaximallyMixedFirstReduction A state has maximally mixed reduction on the first $m$ parties
OpenQuantumProblem35.IsAME A state $\psi$ is absolutely maximally entangled
OpenQuantumProblem35.ExistsAME Existence of an $\mathrm{AME}(n,d)$ state
OpenQuantumProblem35.not_existsAME_zero_dim No absolutely maximally entangled state exists in local dimension $0$ once $n \ge 1$
OpenQuantumProblem35.card_config The number of computational-basis configurations on $m$ parties of local dimension $d$ is $d^m$
OpenQuantumProblem35.maximallyMixed_apply The matrix entries of the maximally mixed state are diagonal and equal to the inverse subsystem dimension
OpenQuantumProblem35.uniformCoeff The common amplitude of the Bell and GHZ witnesses
OpenQuantumProblem35.IsConstantConfig A configuration is constant if all coordinates agree
OpenQuantumProblem35.constantConfig The constant configuration with value $a$
OpenQuantumProblem35.isConstantConfig_constantConfig Every constant configuration is constant
OpenQuantumProblem35.not_isConstantConfig_example A simple binary two-party configuration with different entries is not constant
OpenQuantumProblem35.diagonalState The diagonal $n$-party state: the uniform superposition over constant computational-basis strings
OpenQuantumProblem35.diagonalState_apply Evaluating the diagonal state returns the uniform coefficient on constant strings and 0 otherwise
OpenQuantumProblem35.bellState The standard $d$-dimensional Bell state
OpenQuantumProblem35.ghzState The standard $d$-dimensional GHZ state on $3$ parties
OpenQuantumProblem35.ghzState4 The standard $d$-dimensional GHZ state on $4$ parties
OpenQuantumProblem35.constantCompletion The completion function for constant-support states reduced to one party
OpenQuantumProblem35.constantConfig_injective On a nonempty index type, different constants give different constant configurations
OpenQuantumProblem35.isConstantConfig_iff_exists_constantConfig A configuration on a nonempty index type is constant iff it is equal to some constant configuration
OpenQuantumProblem35.uniformCoeff_norm_sq The squared norm of the uniform coefficient is the inverse local dimension
OpenQuantumProblem35.uniformCoeff_mul_star The squared norm of the uniform coefficient is the inverse local dimension
OpenQuantumProblem35.diagonalState_isNormalized For $n \ge 1$ and $d \ge 1$, the diagonal state is normalized
OpenQuantumProblem35.isConstantConfig_permute_iff Permuting the parties preserves the property of being a constant configuration
OpenQuantumProblem35.diagonalState_permute The diagonal state is invariant under permutations of the parties
OpenQuantumProblem35.constantCompletion_eq_iff A tail configuration equals the constant completion of $x$ iff all of its entries agree with the unique entry of $x$
OpenQuantumProblem35.eq_leftIndex_zero_or_eq_rightIndex Every index in $\mathrm{Fin}\, n$ is either the unique left index or a right index when the left block has size $1$
OpenQuantumProblem35.constantCompletion_injective The completion map for constant configurations is injective once $n \ge 2$
OpenQuantumProblem35.isConstantConfig_combineFirst_one_iff A configuration obtained by combining one entry with a tail is constant iff the tail is the constant completion of that entry
OpenQuantumProblem35.diagonalState_combineFirst_one The diagonal state on a split configuration is nonzero exactly on the graph of the constant completion map
OpenQuantumProblem35.reducedDensityFirst_of_completion A uniform superposition over the graph of an injective completion map has reduced density matrix $(c\overline c) I$ on the first subsystem
OpenQuantumProblem35.hasMaximallyMixedFirstReduction_of_completion The completion criterion gives a maximally mixed reduced state once the coefficient has the correct squared norm
OpenQuantumProblem35.diagonalState_hasMaximallyMixedFirstReduction_one The diagonal state has maximally mixed one-party reductions once $n \ge 2$
OpenQuantumProblem35.diagonalState_isAME_of_div_two_eq_one If $\lfloor n/2 \rfloor = 1$, then the diagonal state is $\mathrm{AME}(n,d)$ for every $d \ge 2$
OpenQuantumProblem35.bellState_isAME The standard Bell state is $\mathrm{AME}(2,d)$ for every physical local dimension $d \ge 2$
OpenQuantumProblem35.ghzState_isAME The standard $3$-party GHZ state is $\mathrm{AME}(3,d)$ for every physical local dimension $d \ge 2$
OpenQuantumProblem35.ame_2_exists The Bell state witnesses the existence of $\mathrm{AME}(2,d)$ for every local dimension $d \ge 2$
OpenQuantumProblem35.ame_3_exists The $3$-party GHZ state witnesses the existence of $\mathrm{AME}(3,d)$ for every local dimension $d \ge 2$
OpenQuantumProblem35.diagonalState_combineFirst_two_of_ne On $4$ parties, the diagonal state vanishes on any split configuration whose first two entries are different
OpenQuantumProblem35.ghzState4_not_ame Sanity check: the standard GHZ family on $4$ parties is not absolutely maximally entangled for any local dimension $d \ge 2$
OpenQuantumProblem35.ame_2_2_exists Source-backed benchmark statement: the Bell state witnesses the existence of an $\mathrm{AME}(2,2)$ state
OpenQuantumProblem35.ame_3_2_exists Source-backed benchmark statement: the three-qubit GHZ state witnesses the existence of an $\mathrm{AME}(3,2)$ state
OpenQuantumProblem35.ame_4_2_not_exists Source-backed benchmark statement: no $\mathrm{AME}(4,2)$ state exists; see Higuchi--Sudbery (2000) and the OQP page
OpenQuantumProblem35.ame_7_2_not_exists Source-backed benchmark statement: no $\mathrm{AME}(7,2)$ state exists; see Huber--Gühne--Siewert (2017) and the OQP page
OpenQuantumProblem35.ame_7_exists_iff For a physical local dimension $d \ge 2$, a seven-party AME state exists exactly when $d \ge 3$
OpenQuantumProblem35.ame_4_6_exists Source-backed benchmark statement: an $\mathrm{AME}(4,6)$ state exists; see Rather et al
OpenQuantumProblem35.ame_7_6_open An $\mathrm{AME}(7,6)$ state exists, by the complete seven-party classification of Shi--Zhang--Zhao--Li (2026)
OpenQuantumProblem35.ame_7_10_open An $\mathrm{AME}(7,10)$ state exists, by the complete seven-party classification of Shi--Zhang--Zhao--Li (2026)
OpenQuantumProblem35.ame_8_4_open Open benchmark statement: does an $\mathrm{AME}(8,4)$ state exist?
OpenQuantumProblem35.ame_8_6_open Open benchmark statement: does an $\mathrm{AME}(8,6)$ state exist?
OpenQuantumProblem35.ame_8_10_open Open benchmark statement: does an $\mathrm{AME}(8,10)$ state exist?
OpenQuantumProblem35.ame_10_6_open Open benchmark statement: does an $\mathrm{AME}(10,6)$ state exist?
OpenQuantumProblem35.ame_10_10_open Open benchmark statement: does an $\mathrm{AME}(10,10)$ state exist?
OpenQuantumProblem35.ame_11_3_open Open benchmark statement: does an $\mathrm{AME}(11,3)$ state exist?
OpenQuantumProblem35.ame_11_6_open Open benchmark statement: does an $\mathrm{AME}(11,6)$ state exist?
OpenQuantumProblem35.ame_12_5_open Does an $\mathrm{AME}(12,5)$ state exist?
OpenQuantumProblem35.ame_12_6_open Open benchmark statement: does an $\mathrm{AME}(12,6)$ state exist?
OpenQuantumProblem35.ame_12_10_open Open benchmark statement: does an $\mathrm{AME}(12,10)$ state exist?
OpenQuantumProblem35.oqp_35 Open Quantum Problem 35: classify all pairs $(n,d)$ with $n \ge 2$ and $d \ge 2$ for which an $\mathrm{AME}(n,d)$ state exists
Constant1a.C1a Tao's Optimization constant 1a / An autocorrelation constant related to Sidon sets: The largest constant
Constant1a.c1a_lower_bound A lower bound, proven by Matolcsi-Vinuesa in [M2010]
Constant1a.c1a_upper_bound An upper bound, proven by Yuksekgonul et al
Constant1a.mem_Ico_c1a How can the best known upper bound $1.5028628587\ldots$ of [T2026] be improved?
Constant1a.mem_Ioc_c1a How can the best known lower bound $1.292$ of [PBV2026] be improved?
Constant1a.c1a_eq What is the exact value of the constant?
BeaverMathOlympiad.beaver_math_olympiad_problem_1 Documented declaration: beaver_math_olympiad_problem_1
BeaverMathOlympiad.beaver_math_olympiad_problem_2_antihydra BMO#2 Antihydra is a sequence starting at 8, and iterating the function
BeaverMathOlympiad.beaver_math_olympiad_problem_2_antihydra.variants.set BMO#2 formulation variant Alternative statement of beaver_math_olympiad_problem_2_antihydra using set size comparison instead of a recurrent sequence b
BeaverMathOlympiad.beaver_math_olympiad_problem_5 Documented declaration: beaver_math_olympiad_problem_5
BeaverMathOlympiad.beaver_math_olympiad_problem_8 Documented declaration: beaver_math_olympiad_problem_8
EquationalTheories_677_255.Equation255_not_implies_Equation677 Equation 255 does not imply Equation 677
EquationalTheories_677_255.Equation677_not_implies_Equation255 Equation 677 does not imply Equation 255
EquationalTheories_677_255.Finite.Equation255_not_implies_Equation677 Note that this is a stronger form of Equation255_not_implies_Equation677
EquationalTheories_677_255.Finite.Equation677_implies_Equation255 Equational Theories, Problem 8.1
Gerstenhaber.finrank_adjoin_pair_le Gerstenhaber's theorem [Ger61]: if $A$ and $B$ are commuting $n \times n$ matrices over a field $K$, then the unital $K$-algebra $K[A, B]$ they generate has dimension at most $n$
Gerstenhaber.finrank_adjoin_triple_le Documented declaration: finrank_adjoin_triple_le
Gerstenhaber.exists_finrank_adjoin_quadruple_gt The analogue of Gerstenhaber's theorem fails for four pairwise commuting matrices: over any field there are four pairwise commuting
HyperbolicGroupsResiduallyFinite.hyperbolic_residuallyFinite Is every hyperbolic group residually finite?
HyperbolicGroupsResiduallyFinite.hyperbolic_exists_finiteIndex_ne_top Does every nontrivial hyperbolic group have a proper subgroup of finite index?
HyperbolicGroupsResiduallyFinite.hyperbolic_residuallyFinite_iff_exists_finiteIndex_ne_top Every hyperbolic group is residually finite if and only if every nontrivial hyperbolic group has a proper subgroup of finite index
Rule30.step The Rule 30 local update on a bi-infinite row row : ℤ → Bool
Rule30.state The Rule 30 evolution from the single-black-cell initial condition: state 0 is black exactly at position $0$, and each subsequent row is obtained by applying step
Rule30.centerColumn The center column of Rule 30: the value of the single central cell (position $0$) at time $t$
Rule30.centerColumn_prefix Sanity check: the first center-column values reproduce the known Rule 30 center column (OEIS A051023)
Rule30.centerColumn_not_eventually_periodic Rule 30 Prize, Problem 1 (non-periodicity)
Rule30.centerColumn_frequency_half Rule 30 Prize, Problem 2 (equal frequency)
SchurTruncatedExponential.truncatedExp Documented declaration: truncatedExp
SchurTruncatedExponential.schur_truncatedExp_galoisGroup_equiv Schur's Theorem (1924): Let f_n(x) = ∑_{j=0}^n x^j/j! be the n-th truncated exponential polynomial over ℚ
SuffixPrefixAvoidance.wordSuffix The suffix of a word w : Fin n → Fin q of length k + 1 (the last k + 1 characters)
SuffixPrefixAvoidance.wordPrefix The prefix of a word w : Fin n → Fin q of length k + 1 (the first k + 1 characters)
SuffixPrefixAvoidance.IsSuffixPrefixAvoiding Two sets of words $A, B$ over Fin q of length n are suffix-prefix avoiding if no nonempty suffix of any word in $A$ equals any prefix of any word in $B$ of the same length
SuffixPrefixAvoidance.words_naive_bound $A$ and $B$ are sets of words of length $n$ over alphabet with $q$ letters
SuffixPrefixAvoidance.suffix_prefix_avoidance_bound $A$ and $B$ are sets of words of length $n$ over alphabet with $q \geq 1$ letters
SymbolLengthMilnorKTheory.symbol_length_problem Documented declaration: symbol_length_problem
SymbolLengthMilnorKTheory.symbol_length_problem.variants.bounded Documented declaration: bounded
SymbolLengthMilnorKTheory.symbol_length_problem.variants.uniform Documented declaration: uniform
SymbolLengthMilnorKTheory.symbol_length_problem.variants.degree_zero Documented declaration: degree_zero
SymbolLengthMilnorKTheory.symbol_length_problem.variants.degree_one Documented declaration: degree_one
SymbolLengthMilnorKTheory.symbol_length_problem.variants.degree_two Documented declaration: degree_two
SymbolLengthMilnorKTheory.symbol_length_problem.variants.degree_two_ge The matching lower bound in degree
SymbolLengthMilnorKTheory.symbol_length_problem.variants.degree_two_isLeast Becher and Hoffmann's question [BecherHoffmann2004, Question 4.6]: is the symbol length of $K^M_2(\mathbb{C}(x_1, \dots, x_m))/p$ exactly $m - 1$?
SymbolLengthMilnorKTheory.symbol_length_problem.variants.zero_mem_of_lt $K^M_n(\mathbb{C}(x_1, \dots, x_m))/p = 0$ for $n > m$: the field has cohomological dimension $m$ [Serre1997, Ch
SymbolLengthMilnorKTheory.symbol_length_problem.variants.isAlgClosed Documented declaration: isAlgClosed
VCDimConvex.hasAddVCDimAtMost_three_of_convex_r2 Every convex set in $\mathbb R^2$ has VC dimension at most 3
VCDimConvex.exists_infinite_convex_r3_shatters There exists a set in $\mathbb R^3$ shattering an infinite set
VCDimConvex.exists_convex_rn_add_two_vc_n_forall_not_hasAddVCNDimAtMost There exists a set of infinite $\mathrm{VC}_n$ dimension in $\mathbb R^{n + 2}$
VCDimConvex.hasAddVCNDimAtMost_two_two_of_convex_r3 Every convex set in $\mathbb R^3$ has $\mathrm{VC}_2$ dimension at most 2
VCDimConvex.exists_exponential_bound_hasAddVCNDimAtMost_n_of_convex_rn_add_one For every $n \ge 1$ there exists some $d$ such that every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most $d$
VCDimConvex.hasAddVCNDimAtMost_n_two_of_convex_rn_add_one Is it true that, for all $n$, every convex set in $\mathbb R^{n + 1}$ has $\mathrm{VC}_n$ dimension at most 3?
AuslanderReiten.subsingleton_ext_of_projective The converse of the conjecture is elementary: over any ring, all higher Ext out of a projective module vanish
AuslanderReiten.auslander_reiten The Auslander-Reiten conjecture [AR75]
AuslanderReiten.auslander_reiten.variants.commutative The Auslander-Reiten conjecture for commutative Noetherian rings [Ki23]
AuslanderReiten.auslander_reiten.variants.local_free Over a Noetherian local ring a finitely generated projective module is free, so the conjecture takes the following form there
AuslanderReiten.auslander_reiten.variants.normal The conjecture holds for normal rings [Ki23, Corollary 1.2]
AuslanderReiten.auslander_reiten.variants.regular_local Over a regular local ring the conjecture is standard: every finitely generated module has finite projective dimension $p$, and if $p > 0$ then $\operatorname{Ext}^p_R(M, R) \ne 0$
BranchingVAS.Bvas A branching vector addition system of dimension d
BranchingVAS.Bvas.Reachable The reachable configurations of a branching vector addition system
BranchingVAS.vecOfList The vector in ℤ^d whose i-th coordinate is the i-th entry of the list l, or 0 if l has fewer than i + 1 entries
BranchingVAS.reachability_decidable The reachability problem for branching vector addition systems is decidable
BranchingVAS.reachable_of_mem_axioms Every axiom that is non-negative is reachable
BranchingVAS.reachable_imp_pos Reachable vectors are always non-negative
BranchingVAS.exampleBvas A small BVAS of dimension 3 used for a test below
BranchingVAS.exampleBvas2 A small (indeed, degenerate, since it contains no binary rules) BVAS of dimension 2 used for a test below
BranchingVAS.reachable_example The vector [0, 0, 12] is reachable in the first example BVAS
BranchingVAS.not_reachable_example The vector [0, 0] is not reachable in the second example BVAS
CardinalityLindelof.HasGδSingletons.lindelof_card Is there a Lindelöf Tychonoff space with singletons as Gδ sets with cardinality greater than the continuum?
CasasAlvero.HasCasasAlveroProp A polynomial P satisfies the Casas-Alvero property if it shares a factor with each of its Hasse derivatives up to order d-1, where d is the degree of P
CasasAlvero.HasCasasAlveroPropᵣ A stronger version of the Casas-Alvero property, which requires that the polynomial P shares a root with each of its Hasse derivatives up to order deg P - 1
CasasAlvero.casas_alvero_iffᵣ Note that whether we use HasCasasAlveroProp or HasCasasAlveroPropᵣ to state the Casas-Alvero conjecture, we obtain the following equivalent statements
CasasAlvero.casas_alvero_conjecture The Casas-Alvero conjecture states that in characteristic zero, if a monic polynomial P has the Casas-Alvero property, then P = (X - α)ᵈ for some α
CasasAlvero.casas_alvero.prime_power The Casas-Alvero conjecture holds for polynomials of prime power degree
CasasAlvero.casas_alvero.double_prime_power The Casas-Alvero conjecture holds for polynomials of degree 2p^k where p is prime
CasasAlvero.casas_alvero.positive_char_counterexample The Casas-Alvero conjecture fails in positive characteristic p for polynomials of degree p + 1
CatchUp.Player An arbitrary two elements type indexing the players in the Catch-Up game
CatchUp.Player.other Returns the other player
CatchUp.Outcome The possible outcomes of a Catch-Up game
CatchUp.Outcome.neg Negates an outcome, swapping win and loss
CatchUp.Outcome.best Computes the best outcome for the current player from a list of possible outcomes
CatchUp.valueAux value remaining s_me s_opp isFirstMove evaluates the position where: * remaining is the set S' of numbers not yet taken
CatchUp.value Public API: The game-theoretic value of Catch-Up on the set S, assuming optimal play
CatchUp.value_of_even_mul_succ_self_div_two Let $T_N = \sum_{k=1}^{N} k = \frac{N(N+1)}{2}$
Chvatal.Decreasing A family F of sets is Decreasing if it is closed under taking subsets
Chvatal.Intersecting A family F of sets is Intersecting if each pair of members has nonempty intersection
Chvatal.exists_maximal_star Documented declaration: exists_maximal_star
ClaudesCycles.Vertex The vertex type: vectors in (ZMod m)³
ClaudesCycles.bumpAt Bump coordinate b of vertex v: add 1 to the b-th component
ClaudesCycles.cubeAdj Adjacency in the cube digraph: u is adjacent to v if v is obtained from u by bumping one coordinate
ClaudesCycles.IsDirectedHamiltonianCycle Documented declaration: IsDirectedHamiltonianCycle
ClaudesCycles.bumpAt_injective The three arcs leaving a vertex are distinct, which is what makes the ∃! in HasHamiltonianArcDecomposition a condition about arcs rather than about heads that might coincide
ClaudesCycles.HasHamiltonianArcDecomposition Documented declaration: HasHamiltonianArcDecomposition
ClaudesCycles.not_hasHamiltonianArcDecomposition_one The hypothesis 1 < m on cube_hamiltonian_arc_decomposition is load-bearing
ClaudesCycles.cube_hamiltonian_arc_decomposition For odd m > 1, the cube digraph on (ZMod m)³ has a Hamiltonian arc decomposition into three directed cycles [Knu26]
ClaudesCycles.cube_hamiltonian_arc_decomposition_impossible_m2 The case m = 2 is impossible: the cube digraph on (ZMod 2)³ does not have a Hamiltonian arc decomposition [Aub82]
ClaudesCycles.cube_hamiltonian_arc_decomposition_even For even m > 2, the cube digraph on (ZMod m)³ has a Hamiltonian arc decomposition
ConjugacyClassSizes.conjClassCard The cardinality of the conjugacy class c
ConjugacyClassSizes.HasDistinctConjClassSizes Documented declaration: HasDistinctConjClassSizes
ConjugacyClassSizes.hasDistinctConjClassSizes_iff Equivalently, two conjugacy classes with the same cardinality must coincide
ConjugacyClassSizes.hasDistinctConjClassSizes_of_subsingleton The trivial group is anti-homogeneous, since it has a single conjugacy class
ConjugacyClassSizes.trivial_center_of_hasDistinctConjClassSizes An anti-homogeneous group has trivial center, since each central element is in its own conjugacy class
ConjugacyClassSizes.hasDistinctConjClassSizes_perm_fin_three The symmetric group $S_3$ is anti-homogeneous, since its three conjugacy classes have sizes $1$, $2$ and $3$
ConjugacyClassSizes.conjClassSizes_iff_sym_three Markel's $S_3$-conjecture (1973): any nontrivial finite ah-group is isomorphic to $S_3$
ConjugacyClassSizes.conjClassSizes_iff_sym_three_solvable The $S_3$-conjecture holds for solvable groups: every nontrivial solvable finite ah-group is isomorphic to $S_3$
DeGiorgi.IsBoundedSolution The function $u : ℝ^n → ℝ$ is a bounded classical solution to $Δ u + u - u^3 = 0$
DeGiorgi.HasPositiveDeriv The first partial derivative of $u : ℝ^n → ℝ$ is strictly positive
DeGiorgi.HasHyperplaneLevelSets The level sets of $u : ℝ^n → ℝ$ are hyperplanes
DeGiorgi.DeGiorgi_conclusion Documented declaration: DeGiorgi_conclusion
DeGiorgi.DeGiorgi_le_eight De Giorgi's conjecture holds in dimension $n ≤ 8$
DeGiorgi.DeGiorgi_ge_nine In dimension $n ≥ 9$, the conclusion of De Giorgi's conjecture does not hold
DeGiorgi.DeGiorgi_one De Giorgi's conjecture trivially holds in dimension $n = 1$
DeGiorgi.DeGiorgi_two De Giorgi's conjecture holds in dimension $n = 2$
DeGiorgi.DeGiorgi_three De Giorgi's conjecture holds in dimension $n = 3$
DeGiorgi.DeGiorgi_four De Giorgi's conjecture holds in dimension $n = 4$
DeGiorgi.DeGiorgi_five De Giorgi's conjecture holds in dimension $n = 5$
DeGiorgi.DeGiorgi_six De Giorgi's conjecture holds in dimension $n = 6$
DeGiorgi.DeGiorgi_seven De Giorgi's conjecture holds in dimension $n = 7$
DeGiorgi.DeGiorgi_eight De Giorgi's conjecture holds in dimension $n = 8$
DegreeSequencesTriangleFree.IsCompactSequenceOn A sequence of natural numbers is compact on a set S if consecutive terms at distance 2 differ by 1 for all k ∈ S
SimpleGraph.degreeFreq The number of vertices of G having degree d
DegreeSequencesTriangleFree.lemma1_a Lemma 1 (a) If a sequence d is nondecreasing and no three terms are equal, then terms at distance 2 differ by at least 1
DegreeSequencesTriangleFree.lemma1_b Lemma 1 (b) If a sequence d is nondecreasing and no three terms are equal, then terms at distance 2 * r differ by at least r
DegreeSequencesTriangleFree.lemma2_helper_short Helper: additive form of Lemma 2(a)'s estimate, used by lemma2_a–lemma2_d
DegreeSequencesTriangleFree.lemma2_a Lemma 2 (a) Inequality involving sums of terms of a nondecreasing sequence with no three terms equal
DegreeSequencesTriangleFree.lemma2_b Lemma 2 (b) Inequality involving sums of terms of a nondecreasing sequence with no three terms equal
DegreeSequencesTriangleFree.lemma2_helper_shift Documented declaration: lemma2_helper_shift
DegreeSequencesTriangleFree.lemma2_c Lemma 2 (c) Inequality involving sums of terms of a nondecreasing sequence with no three terms equal
DegreeSequencesTriangleFree.lemma2_d Lemma 2 (d) Inequality involving sums of terms of a nondecreasing sequence with no three terms equal
SimpleGraph.HasCompactdegreeSequence The degree sequence of G is compact if it satisfies IsCompactSequenceOn for all valid indices k such that k + 2 < Fintype.card α
SimpleGraph.theorem1 Theorem 1
SimpleGraph.lemma3 Lemma 3
SimpleGraph.lemma4 Lemma 4
SimpleGraph.theorem2 Theorem 2
SimpleGraph.F F n is the smallest number of vertices of a triangle-free graph with chromatic number n and f = 3
SimpleGraph.F_three The smallest number of vertices of a triangle-free graph with chromatic number 3 and f=3 is 7
SimpleGraph.F_four_le The smallest number of vertices of a triangle-free graph with chromatic number 4 and f=3 is at most 19
DubnerConjecture.IsTwinPrime A twin prime is a prime number that has a prime gap of 2, meaning either p - 2 or p + 2 is also prime
DubnerConjecture.dubner_conjecture Every even number greater than 4208 is the sum of two twin primes
EisenbudGreenHarris.eisenbud_green_harris The Eisenbud-Green-Harris conjecture
EisenbudGreenHarris.eisenbud_green_harris.variants.macaulay Macaulay's theorem: every homogeneous ideal has the same Hilbert function as a lex ideal
EisenbudGreenHarris.eisenbud_green_harris.variants.clements_lindstrom The Clements-Lindström theorem [CL69]: the conjecture holds for an ideal that already contains the pure powers $x_1^{d_1}, \dots, x_c^{d_c}$
EisenbudGreenHarris.eisenbud_green_harris.variants.caviglia_deStefani The conjecture holds when the degrees grow quickly enough, namely when $d_i \ge \sum_{j < i} (d_j - 1)$ for all $i \ge 3$
FusibleNumber.IsFusible A rational number is fusible if it belongs to the smallest set containing $0$ and closed under the operation $$ a \sim b = \frac{a + b + 1}{2} $$ whenever $|a-b| < 1$
FusibleNumber.isFusible_one_half The rational number $1/2$ is fusible
FusibleNumber.isFusible_one The rational number $1$ is fusible
FusibleNumber.conj_7_1 Documented declaration: conj_7_1
Gourevitch.gourevitch_series_identity The Gourevitch series identity: The following identity holds
AlgebraicGeometry.Scheme.VectorBundles A vector bundle over a scheme S is a locally free $\mathcal{O}_S$-module of finite rank
AlgebraicGeometry.Scheme.VectorBundles.Splitting A splitting of a vector bundle 𝓕 is a non-trivial direct sum decomposition of 𝓕
AlgebraicGeometry.Scheme.harthshorne_conjecture There are no indecomposable vector bundles of rank 2 on $\mathbb{P}^n$ for $n \ge 7$
Homogeneous.HomogeneousSpace A topological space $X$ is called homogeneous if for all $x, y \in X$ there is homeomorphism $f : X \to X$ with $f(x) = y$
Homogeneous.DiscreteTopology.toHomogeneousSpace Every discrete space is homogeneous
Homogeneous.homogeneousSpace_exists_inj_tendsto Problem 13 in [Ar2013]: Is it true that every infinite homogeneous compact hausdorff space contains a non-trivial convergent sequence?
Homogeneous.homogeneousSpace_exists_surjective Problem 14 in [Ar2013]: Is it possible to represent an arbitrary compact hausdorff space as an image of a homogeneous compact space under a continuous mapping?
Homogeneous.CountablyMonolithicSpace A topological space is called ω-monolithic if the closure of every countable subspace is metrizable
Homogeneous.MetrizableSpace.countablyMonolithicSpace Every Metrizable space is ω-monolithic
Homogeneous.firstCountableTopology_of_countablyMonolithicSpace Problem 15 in [Ar2013]: Is every homogeneous ω-monolithic compact hausdorff space first countable?
Homogeneous.countablyMonolithicSpace_card_lt Problem 16 in [Ar2013]: Is the cardinality of every homogeneous ω-monolithic compact hausdorff space not greater than 𝔠?
Homogeneous.MonolithicSpace A topological space is monolithic (see [Ar1987]; [Ar2013] uses the term without defining it) if every subspace has network weight at most its density
Homogeneous.monolithicSpace_iff_networkWeight_closure_le Monolithicity in the form nw(cl A) ≤ |A| + ω for every subset A
Homogeneous.DiscreteTopology.toMonolithicSpace Every discrete space is monolithic
Homogeneous.SecondCountableTopology.toMonolithicSpace Every second countable space is monolithic
Homogeneous.monolithicSpace_exists_nhds_generated_countable Problem 17 in [Ar2013]: Is it true that every nonempty monolithic compact hausdorff space contains a point with a first countable neighborhood basis?
KaplanskyUInvariant.IsUInvariant IsUInvariant n means that some field of characteristic not 2 has $u$-invariant $n$, i.e. $n$ is the largest dimension of an anisotropic quadratic form over that field
KaplanskyUInvariant.isGreatest_anisotropicDims_complex $u(\mathbb{C}) = 1$ [MerkurjevParimala2025, §5.1]: the form $x^2$ is anisotropic, and every quadratic form in two or more variables over an algebraically closed field is isotropic
KaplanskyUInvariant.not_bddAbove_anisotropicDims_real $u(\mathbb{R}) = \infty$ [MerkurjevParimala2025, §5.1]: the sum of $n$ squares is anisotropic for every $n$, so there is no largest dimension of an anisotropic form
KaplanskyUInvariant.u_invariant_values The values of the $u$-invariant, stated as in [MerkurjevParimala2025, §5.1]: which integers are $u$-invariants of fields (of characteristic not $2$)?
KaplanskyUInvariant.u_invariant_values.variants.kaplansky_conjecture Kaplansky's conjecture [Kaplansky1953, p
KaplanskyUInvariant.u_invariant_values.variants.not_three The $u$-invariant is never $3$ [Kaplansky1953, Theorem 2]; see also [Lam2005, Proposition XI.6.8] and [EKM2008, Corollary 36.4]
KaplanskyUInvariant.u_invariant_values.variants.not_five The $u$-invariant is never $5$ [Lam2005, Proposition XI.6.8], [EKM2008, Corollary 36.4]
KaplanskyUInvariant.u_invariant_values.variants.not_seven The $u$-invariant is never $7$ [Lam2005, Proposition XI.6.8], [EKM2008, Corollary 36.4]
KaplanskyUInvariant.u_invariant_values.variants.two_pow Every power of $2$ is a $u$-invariant [Kaplansky1953, p
KaplanskyUInvariant.u_invariant_values.variants.even Merkurjev's theorem [Merkurjev1991], see also [EKM2008, Theorem 38.4]: every positive even integer is a $u$-invariant
KaplanskyUInvariant.u_invariant_values.variants.nine Izhboldin's theorem [Izhboldin2001]: there is a field of $u$-invariant $9$, the first example of an odd $u$-invariant greater than $1$
KaplanskyUInvariant.u_invariant_values.variants.two_pow_add_one Vishik's theorem [Vishik2009, Corollary 5.2]: for every $r \ge 3$ there is a field of $u$-invariant $2^r + 1$
KaplanskyUInvariant.u_invariant_values.variants.karpenko Karpenko's theorem [Karpenko2026, Theorem 1.1]: if neither $n + 1$ nor $n + 3$ is a power of $2$, then there is a field of $u$-invariant $n$
KaplanskyUInvariant.u_invariant_values.variants.eleven The special case
KaplanskyUInvariant.u_invariant_values.variants.odd The expectation recorded in [MerkurjevParimala2025, §5.1]: every odd integer $\ge 9$ is a $u$-invariant
KaplanskyUInvariant.u_invariant_values.variants.thirteen The smallest open case of the form $2^r - 3$: is there a field of $u$-invariant $13$?
KaplanskyUInvariant.u_invariant_values.variants.fifteen The smallest open case of the form $2^r - 1$: is there a field of $u$-invariant $15$?
KotzigConjecture.kotzig_conjecture For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of $T$ via cyclic shifts of a single embedding
Kurepa.left_factorial Left factorial of n $$!n = 0! + 1! + 2! + \dots + (n-1)!$$
Kurepa.kurepa_conjecture Kurepa's conjecture For all $n$, $$!n\not\equiv 0 \mod n$$ This appears as B44 "Sums of factorials." in Unsolved Problems in Number Theory by *Richard K
Kurepa.kurepa_conjecture.variants.prime This statement can be reduced to the prime case only
Kurepa.kurepa_conjecture.prime_reduction Kurepa's conjecture for all integers greater than 2 is equivalent to the conjecture restricted to primes greater than 2
Kurepa.kurepa_conjecture.variants.gcd An equivalent formulation in terms of the gcd of $n!$ and $!n$
Kurepa.kurepa_conjecture.gcd_reduction Kurepa's conjecture for all integers greater than 2 is equivalent to the statement that $\gcd(n!, !n) = 2$ for all integers greater than 2
Kurepa.kurepa_conjecture.variants.first_cases Sanity check: for small values we can just compute that the conjecture is true
Kurepa.kurepa_conjecture.variants.gcd.first_cases Sanity check: for small values we can just compute that the conjecture is true
LatinSquare.Orthogonal Two latin squares of the same order are orthogonal if superimposing them gives each ordered pair of symbols at most once
LatinSquare.MutuallyOrthogonal A family of latin squares is mutually orthogonal if any two distinct members are orthogonal
LatinSquare.HasCompleteMOLS A complete set of mutually orthogonal latin squares (MOLS) of order n consists of n - 1 latin squares of order n, pairwise orthogonal to each other
LatinSquare.oddOrderLatinSquareTransversal Conjecture 3.2 in [Wa2011]: Each Latin square of odd order has at least one transversal
LatinSquare.oddOrderLeq9LatinSquareTransversal The conjecture is known to be true for $n \leq 9$
LatinSquare.latinSquareOrder11Transversal The smallest odd number for which this conjecture is not known is 11
LatinSquare.latinSquareNearTransversal Conjecture 5.1 in [Wa2011]: Every latin square has a near-transversal
LatinSquare.z The number of transversals of the Cayley table of the cyclic group $\mathbb{Z}_n$
LatinSquare.z_zero The $0 \times 0$ Cayley table has exactly $1$ transversal (vacuously)
LatinSquare.z_odd_values The number of transversals of the Cayley table of $\mathbb{Z}_n$ for odd $n$ forms OEIS A006717, starting with $z(1) = 1, z(3) = 3, z(5) = 15, z(7) = 133$
LatinSquare.z_even The Cayley table of $\mathbb{Z}_n$ for positive even $n$ has no transversals
LatinSquare.numTransversalsZn Conjecture 6.7 in [Wa2011]: There exist real constants $0 < c_1 < c_2 < 1$ such that $$ c_1^n n! \leq z_n \leq c_2^n n! $$ for all odd $n \geq 3$
LatinSquare.growthRateZn Conjecture 6.9 in [Wa2011]: $$ \lim_{\substack{n \to \infty \\ n \text{ odd}}} \frac{1}{n} \log(z_n / n!) = -1 $$ It is not even known if this limit exists
LatinSquare.T The maximum number of transversals over all latin squares of order n
LatinSquare.maxTransversalsBound Theorem 7.2 in [Wa2011]: For all $n \geq 5$, $$ 15^{n/5} \leq T(n) \leq c^n \sqrt{n} \cdot n! $$ where $c = \sqrt{\frac{3 - \sqrt{3}}{6}} \cdot e^{\sqrt{3}/6}$
LatinSquare.molsExistenceProblem MOLS existence problem: determine exactly which orders n admit a complete set of n - 1 mutually orthogonal latin squares
LatinSquare.molsOrder12 The smallest unresolved case of the MOLS existence problem: whether there are 11 mutually orthogonal latin squares of order 12
LatinTableau.SimpleGraph.LatinTableauConjecture The Latin Tableau Conjecture: If G is the simple graph of a Young diagram, then G is CDS-colorable
MonochromaticQuantumGraph.V Vertices of $K_N$
MonochromaticQuantumGraph.EdgeN Edge label for $K_N$ with endpoint indices in Fin D
MonochromaticQuantumGraph.WeightsN Weights on edges
MonochromaticQuantumGraph.mkEdge Helper: build an EdgeN from endpoints and endpoint indices
MonochromaticQuantumGraph.vertices Ordered vertex list $[0, 1, \ldots, N-1]$
MonochromaticQuantumGraph.allEqualList Chain-equality along a list of vertices
MonochromaticQuantumGraph.allEqual All indices equal on Fin N (using the canonical ordered vertex list)
MonochromaticQuantumGraph.pmSumListAux Auxiliary perfect-matching sum on a vertex list, using a fuel parameter n for termination
MonochromaticQuantumGraph.pmSumList Perfect-matching sum on a list: run pmSumListAux with fuel = L.length
MonochromaticQuantumGraph.pmSumN The perfect-matching sum for $K_N$: use the canonical ordered vertex list vertices N
MonochromaticQuantumGraph.EqSystemN The monochromatic quantum graph equation system for $K_N$
MonochromaticQuantumGraph.instDecidableEqSystemN Instance: EqSystemN N D W is decidable when α has decidable equality
MonochromaticQuantumGraph.eqSystem4_d2_nat Sanity check over ℕ using native_decide
MonochromaticQuantumGraph.eqSystem4_d3_nat Sanity check over ℕ using native_decide
MonochromaticQuantumGraph.eqSystem6_d2_nat Sanity check over ℕ using native_decide
MonochromaticQuantumGraph.eqSystem4_no_solution_nnreal_ge4 Bogdanov: for $N = 4$ and all $D \geq 4$, no solution exists over $\mathbb{R}_{\geq 0}$
MonochromaticQuantumGraph.eqSystem_no_solution_nnreal_even_ge6_ge3 Bogdanov: for all even $N \geq 6$ and $D \geq 3$, no solution exists over $\mathbb{R}_{\geq 0}$
MonochromaticQuantumGraph.eqSystem6_no_solution_ge3 For $N = 6$ and all $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem8_no_solution_d3 For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem10_no_solution_d3 For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem10_no_solution_d4 For $N = 10$ and $D = 4$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem10_no_solution_d5 For $N = 10$ and $D = 5$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem10_no_solution_d6 For $N = 10$ and $D = 6$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem10_no_solution_d7 For $N = 10$ and $D = 7$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem12_no_solution_d3 For $N = 12$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem14_no_solution_d3 For $N = 14$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem16_no_solution_d3 For $N = 16$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem_no_solution_ge6_ge3 For all even $N \geq 6$ and $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{C}$?
MonochromaticQuantumGraph.eqSystem6_no_solution_d3_real For $N = 6$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{R}$?
MonochromaticQuantumGraph.eqSystem6_no_solution_ge3_real For $N = 6$ and all $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{R}$?
MonochromaticQuantumGraph.eqSystem8_no_solution_d3_real For $N = 8$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{R}$?
MonochromaticQuantumGraph.eqSystem10_no_solution_d3_real For $N = 10$ and $D = 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{R}$?
MonochromaticQuantumGraph.eqSystem_no_solution_ge6_ge3_real For all even $N \geq 6$ and $D \geq 3$, does there exist no solution to the monochromatic quantum graph equation system over $\mathbb{R}$?
NLSCritical.LocalSchwartzSolution A smooth solution to the NLS equation on the time interval $[-l, l]$
NLSCritical.energy The Hamiltonian of the energy-critical NLS in $ℝ^3$
NLSCritical.L10_local The space-time $L^10$ norm, restricted to the time interval $[-l,l]$
NLSCritical.NLS_apriori The $L^10_{t,x}$ a priori estimate for local schwartz solutions
PfisterPythagorasNumber.isLeast_pythagorasBounds_real Every sum of squares in $\mathbb{R}$ is a square and $1$ is not a sum of zero squares, so $p(\mathbb{R}) = 1$
PfisterPythagorasNumber.pfister_problem Pfister's problem (Problem 1 of [Pfister1971, §4]): what is the true value of $p(\mathbb{R}(X_1, \dots, X_n))$, as a function of $n$?
PfisterPythagorasNumber.pfister_problem.variants.eq_two_pow Pfister's problem in its common yes/no form (e.g. Question 0.2 of [Benoist2017]): is $p(\mathbb{R}(X_1, \dots, X_n)) = 2^n$ for every $n$, i.e. is Pfister's bound optimal?
PfisterPythagorasNumber.pfister_problem.variants.upper_bound Pfister's theorem: every sum of squares in $\mathbb{R}(X_1, \dots, X_n)$ is a sum of $2^n$ squares [Pfister1995, p
PfisterPythagorasNumber.pfister_problem.variants.cassels Cassels' theorem [Cassels1964], as quoted in Problem 1 of [Pfister1971, §4]: $1 + X_1^2 + \dots + X_n^2$ is not a sum of $n$ squares in $\mathbb{R}(X_1, \dots, X_n)$
PfisterPythagorasNumber.pfister_problem.variants.add_one_le The lower bound
PfisterPythagorasNumber.pfister_problem.variants.lower_bound The lower bound $n + 2 \le p(\mathbb{R}(X_1, \dots, X_n))$ for $n \ge 2$ [Pfister1995, p
PfisterPythagorasNumber.pfister_problem.variants.zero Documented declaration: zero
PfisterPythagorasNumber.pfister_problem.variants.one $p(\mathbb{R}(X)) = 2$ [Pfister1995, p
PfisterPythagorasNumber.pfister_problem.variants.two $p(\mathbb{R}(X_1, X_2)) = 4$ [CEP1971], as recorded in [Pfister1971, §4] and [Pfister1995, p
PrimeTuplesConjecture.prime_tuples_conjecture For any k ≥ 2, let a₁,...,aₖ and b₁,...,bₖ be integers with aᵢ > 0
ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture For a graph $G$, we define $\Delta(G)$ to be the maximum degree, $\omega(G)$ to be the size of the largest clique subgraph, and $\chi(G)$ to be the chromatic number
ReedOmegaDeltaChi.reed_omega_delta_chi_conjecture_for_finite_graphs For a finite graph $G$, we define $\Delta(G)$ to be the maximum degree, $\omega(G)$ to be the size of the largest clique subgraph, and $\chi(G)$ to be the chromatic number
ReedOmegaDeltaChi.reed_conjecture_Δ_6_ω_2 The simplest open case is when $\Delta(G) = 6$ and $\omega(G) = 2$
RiemannHypothesis.robin_criterion Robin's Criterion
RiemannHypothesis.robin_criterion_rhs The statement that Robin's criterion holds
RiemannHypothesis.lagarias_criterion Lagarias' Criterion
RiemannHypothesis.lagarias_criterion_rhs The statement that Lagarias' criterion holds
RiemannHypothesis.nontrivialPosZeros The nontrivial zeros of the Riemann zeta function with positive imaginary part
RiemannHypothesis.lambda Documented declaration: lambda
RiemannHypothesis.li_criterion Li's Criterion
RiemannHypothesis.li_criterion_rhs The statement that Li's criterion holds
RiemannHypothesis.fractScalarInv The function $\rho_{\theta} := \left\{\frac{\theta}{x}\right\}$ appearing in the Beurling-Nyman criterion
RiemannHypothesis.fractScalarInvSumL2 The linear combination $\sum_{\nu = 1}^n c_{\nu}\rho_{\theta_\nu}$ expressed explicitly as an element of $L^2(0, 1)$
RiemannHypothesis.beurling_nyman_criterion Beurling-Nyman Criterion
RiemannHypothesis.beurling_nyman_criterion_rhs The statement that the Beurling-Nyman criterion holds
RiemannHypothesis.weilArchimedean The additive Archimedean factor appearing in the Weil criterion
RiemannHypothesis.weilNonArchimedean The additive non-Archimedean factor appearing in the Weil criterion
RiemannHypothesis.IsTestFunction The additive condition on test functions
RiemannHypothesis.weil_positivity_criterion Weil's Positivity Criterion
RiemannHypothesis.weil_positivity_criterion_rhs The statement that the Weil criterion holds
RiemannZetaMoments.moment The $k$th moment of the Riemann zeta function is the integral $$ \frac{1}{T}\int_0^T |\zeta(1/2 + it)|^{2k}dt. $$
RiemannZetaMoments.moment_isEquiv Documented declaration: moment_isEquiv
RiemannZetaMoments.moment_isEquiv₁ The asymptotic behaviour of the first moment of the Riemann zeta function was proved by Hardy and Littlewood, where $c_1 = 1$ [Co26, §3.4.5]
RiemannZetaMoments.moment_isEquiv₂ The asymptotic behaviour of the second moment of the Riemann zeta function was proved by Ingham [In26], where $c_2 = 1/(2\pi^2)$
RiemannZetaMoments.arithmeticFactor Documented declaration: arithmeticFactor
RiemannZetaMoments.moment_isEquiv₃ The constant in the asymptotic formula for the 3rd zeta moment is conjectured to be $42a_3/9!$ [CG98]
RiemannZetaMoments.moment_isEquiv₄ The constant in the asymptotic formula for the 4th zeta moment is conjectured to be $24024a_4/16!$ [CGo01]
RiemannZetaMoments.moment_isEquivₖ The Keating–Snaith conjecture [KS00] for the $k$th moment of the Riemann zeta function, provides the value of $c_k$ in the asymptotic formula [Co26, §3.4.5]
RingelConjecture.ringel_conjecture For any tree $T$ with $n$ edges, the complete graph $K_{2n+1}$ decomposes into $2n+1$ edge-disjoint copies of $T$
Rupert.transformed_shadow The result of transforming a subset of ℝ³ by a chosen rotation and offset, and then projected to ℝ²
Rupert.IsRupert Documented declaration: IsRupert
Rupert.is_every_convex_polyhedron_rupert There exists a convex polyhedron with nonempty interior for which the Rupert property does not hold
SerreUniformity.cmJInvariants The thirteen rational CM $j$-invariants
SerreUniformity.HasFullTorsionAction Every additive automorphism of $E[p]$ is induced by a Galois automorphism
SerreUniformity.serre_uniformity Serre's uniformity question over
SerreUniformity.serre_uniformity.variants.bound_37 Serre's uniformity conjecture over
StrongSensitivityConjecture.flip Flip operator, flip x B returns input x with bits in block B inverted
StrongSensitivityConjecture.sensitivityAt Local sensitivity s(f,x), number of indices where flipping one bit changes the value of f
StrongSensitivityConjecture.sensitivity Global sensitivity s(f), maximum sensitivity of f over all inputs
StrongSensitivityConjecture.IsValidBlockConfig Check validity of block collection (disjoint and sensitive), A collection of blocks cB is valid for f at x if the blocks are disjoint and flipping any block changes f(x)
StrongSensitivityConjecture.blockSensitivityAt Local block sensitivity bs(f,x), maximum size of a collection of sensitive, disjoint blocks for f at x
StrongSensitivityConjecture.blockSensitivity Global block sensitivity of f, maximum block sensitivity of f over all inputs
StrongSensitivityConjecture.strong_sensitivity_conjecture Strong Sensitivity Conjecture, for every Boolean function f : {0,1}^n → {0,1}, bs(f) ≤ s(f)^2
StrongSensitivityConjecture.nisanExample Simple test example
StrongSensitivityConjecture.nisanExample_sensitivity Assuming n is a positive multiple of 4, the sensitivity of nisanExample is $n/2 + 2$
StrongSensitivityConjecture.nisanExample_blockSensitivity Assuming n is a multiple of 4 and $n \ge 8$, the block sensitivity of nisanExample is $3n/4$, achieved by any $x$ with Hamming weight $n/2$
TuDengConjecture.binaryWeight The binary weight of a natural number: the number of ones in its binary expansion
TuDengConjecture.tu_deng_conjecture The Tu-Deng conjecture
VoronovskajaTypeFormula.bernsteinTail Cumulative sum $J_{n,k}(x) = \sum_{j=k}^n p_{n,j}(x)$
VoronovskajaTypeFormula.bezierBernstein Bézier–type Bernstein operator: $$ (B_{n,\alpha} f)(x) = \sum_{k=0}^{n} f\!\left(\frac{k}{n}\right) \left( J_{n,k}(x)^{\alpha} - J_{n,k+1}(x)^{\alpha} \right) $$
VoronovskajaTypeFormula.bernsteinTail_zero The zeroth Bernstein tail is the constant polynomial one
VoronovskajaTypeFormula.bernsteinTail_succ_self The Bernstein tail after the final index vanishes
VoronovskajaTypeFormula.bernsteinTail_eval_eq_binomial_tail Evaluation identifies bernsteinTail with the usual binomial tail polynomial
VoronovskajaTypeFormula.bezierBernstein_eq_sum_binomial_tail The Bézier--Bernstein operator written directly in terms of the explicit binomial tails $J_{n,k}(x) = \sum_{j=k}^{n}\binom{n}{j}x^{j}(1-x)^{n-j}$
VoronovskajaTypeFormula.bernsteinTail_eval_antitone Bernstein tails decrease as their lower index increases
VoronovskajaTypeFormula.bezier_weight_nonneg Every Bézier--Bernstein coefficient is nonnegative on the unit interval
VoronovskajaTypeFormula.sum_bezier_weights The Bézier--Bernstein coefficients telescope to one
VoronovskajaTypeFormula.bezierBernstein_const Bézier--Bernstein operators reproduce constant functions
VoronovskajaTypeFormula.bezierBernstein_id On the identity function, the operator is the normalized sum of its powered binomial tails
VoronovskajaTypeFormula.eventually_scaled_bezierBernstein_id For the identity function, the scaled approximation error is eventually exactly the centered, powered-binomial-tail sum
VoronovskajaTypeFormula.bezierBernstein_zero Bézier--Bernstein operators evaluate exactly at the left endpoint
VoronovskajaTypeFormula.bezierBernstein_one Bézier--Bernstein operators evaluate exactly at the right endpoint for positive degree
VoronovskajaTypeFormula.bezierBias The proposed first-order bias constant for the Bézier--Bernstein operator
VoronovskajaTypeFormula.one_sub_cdf_gaussianReal_le_exp Chernoff bound for the standard Gaussian: $1 - \Phi(t) \le e^{-t^2/2}$ for $t \ge 0$
VoronovskajaTypeFormula.integrableOn_one_sub_cdf_gaussianReal_rpow The positive-tail part of the bias integrand is integrable on $(0, \infty)$
VoronovskajaTypeFormula.integrableOn_one_sub_cdf_gaussianReal_pow The negative-tail part of the bias integrand is integrable on $(0, \infty)$
VoronovskajaTypeFormula.bezierBias_eq_sub bezierBias is the difference of the two tail integrals $\int_0^\infty (1 - \Phi(t))^\alpha\,dt$ and $\int_0^\infty (1 - \Phi(t)^\alpha)\,dt$
VoronovskajaTypeFormula.bezierBias_mul_sqrt_mul_iteratedDerivWithin On the unit interval the limit value may equivalently be written with deriv instead of iteratedDerivWithin: at the endpoints the factor
VoronovskajaTypeFormula.voronovskaja_theorem.bernstein_operators Classical Voronovskaja theorem (α = 1)
VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.boundary The proposed asymptotic formula holds unconditionally at the two endpoints of the unit interval
VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth Version of the asymptotic formula which treats "sufficiently smooth" as an eventual condition in the smoothness order
VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.eventually_smooth.limit_exists Existence-only version of the eventual-smoothness variant
VoronovskajaTypeFormula.voronovskaja_theorem.bezier_bernstein_operators.variants.answer_smoothness Version of the asymptotic formula which records the sufficient smoothness order together with the limit: the order $2$ and the formula $\mu_\alpha\sqrt{x(1-x)}\,f'(x)$ work
WeakTiling.IsUnionOfNIntervals Documented declaration: IsUnionOfNIntervals
WeakTiling.IsFiniteUnionOfIntervals A set $A \subseteq \mathbb{R}$ is a finite union of intervals if it is a union of $n$ intervals for some $n$
WeakTiling.IsWeakTilingMeasure A positive, locally finite Borel measure
WeakTiling.IsProperTiling Documented declaration: IsProperTiling
WeakTiling.HasBoundedDensity Documented declaration: HasBoundedDensity
WeakTiling.problem_4_2 Problem 4.2
WeaklyFirstCountable.WeaklyFirstCountableTopology A topological space
WeaklyFirstCountable.IsCellularFamily A cellular family is a pairwise-disjoint collection of nonempty open sets
WeaklyFirstCountable.souslinNumber The Souslin number of a topological space is the supremum of the cardinalities of its cellular families
WeaklyFirstCountable.HasCountableSouslinNumber A space has countable Souslin number when its Souslin number is at most ℵ₀
WeaklyFirstCountable.exists_weakly_first_countable_not_first_countable There are weakly first countable spaces which are not first countable, for example the Arens Space
WeaklyFirstCountable.FirstCountableTopology.weaklyFirstCountableTopology Every first countable space is weakly first countable, simply take $N x$ as a countable neighborhood basis of $x$
WeaklyFirstCountable.hasCountableSouslinNumber_of_separable Every separable space has countable Souslin number
WeaklyFirstCountable.existsWeaklyFirstCountableCompactBig Problem 2 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space X such that $𝔠 < |X|$
WeaklyFirstCountable.existsWeaklyFirstCountableCompactNotFirstCountable Problem 3 in [Ar2013]: Give an example in ZFC of a weakly first- countable compact Hausdorff space which is not first countable
WeaklyFirstCountable.CH.existsWeaklyFirstCountableCompactNotFirstCountable Under CH, such a space (for Problem 3 in [Ar2013]) exists as constructed in [Ya1976] by Yakovlev
WeaklyFirstCountable.cardinalMk_le_continuum_of_weaklyFirstCountable_of_countableSouslinNumber Problem 4 in [Ar2013]: If a Tychonoff weakly first-countable space has countable Souslin number, then does its cardinality not exceed the continuum?
ZagierMZV.multiZeta The multiple zeta value
ZagierMZV.weight The weight of an MZV index $(s_1, \ldots, s_k)$ is $s_1 + \cdots + s_k$
ZagierMZV.AdmissibleIndex An MZV index is admissible if it is either empty or if the first entry is at least 2 and all entries are positive
ZagierMZV.mzvSetOfWeight The set of all MZV values of weight $n$
ZagierMZV.mzvSpanOfWeight The $\mathbb{Q}$-submodule of $\mathbb{R}$ spanned by all MZVs of weight $n$
ZagierMZV.zagierDim The conjectured Zagier dimension sequence $d_n$, defined by $d_0 = 1$, $d_1 = 0$, $d_2 = 1$, and $d_n = d_{n-2} + d_{n-3}$ for $n \geq 3$
ZagierMZV.zagier_conjecture Documented declaration: zagier_conjecture
ZagierMZV.zagier_upper_bound Upper bound [Te02, DG05] The dimension of the $\mathbb{Q}$-vector space of MZVs of weight $n$ is at most $d_n$
ZagierMZV.zagierDim_first_values The first few values of $d_n$ are $1, 0, 1, 1, 1, 2, 2, 3, 4, 5, 7, 9, \ldots$
ZagierMZV.no_admissible_weight_one There is no admissible index of weight 1 (since $s_1 \geq 2$ is required)
ZagierMZV.dim_mzv_weight_zero $\mathcal{Z}_0 = \mathbb{Q}$, so $\dim_\mathbb{Q}(\mathcal{Z}_0) = 1$
ZagierMZV.dim_mzv_weight_one $\mathcal{Z}_1 = \emptyset$, so $\dim_\mathbb{Q}(\mathcal{Z}_1) = 0$
ZagierMZV.multiZeta_two Euler's identity: $\zeta(2) = \pi^2/6$
ZagierMZV.multiZeta_four Euler's identity for $\zeta(4) = \pi^4/90$
ABC.radical The radical of n denoted is the product of the distinct prime factors of n
ABC.quality Quality q(a, b, c) of the triple (a, b, c) is defined as q(a,b,c) = log (c) / log (rad(abc))
ABC.abc For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers, with a + b = c, such that c > rad(abc)^(1+ε)
ABC.abc.variants.lt_constant_mul For every positive real number ε, there exists a constant K_ε such that for all triples (a, b, c) of coprime positive integers, with a + b = c we have c < K_ε rad(abc)^(1+ε)
ABC.abc.variants.quality For every positive real number ε, there exist only finitely many triples (a, b, c) of coprime positive integers with a + b = c such that q(a, b, c) > 1 + ε
AgohGiuga.AgohGiugaCongr The Agoh-Giuga Conjecture, Agoh's formulation
AgohGiuga.AgohGiugaSum The Agoh-Giuga Conjecture, Giuga's formulation
AgohGiuga.agoh_giuga The Agoh-Giuga Conjecture, Agoh's formulation
AgohGiuga.agoh_giuga.variants.giuga The Agoh-Giuga Conjecture, Giuga's formulation
AgohGiuga.agoh_giuga.variants.equivalence The two statements of the conjecture are equivalent
AgohGiuga.IsWeakGiuga A (weak) Giuga number is a composite number $n$ such that $$\sum_{i=1}^{n - 1}i^{\varphi(n)} \equiv -1\pmod{n}$$
AgohGiuga.IsStrongGiuga A (strong) Giuga number is a composite number $n$ such that $$\sum_{i=1}^{n - 1}i^{n - 1} \equiv -1\pmod{n}$$
AgohGiuga.isWeakGiuga_iff_sum_primeFactors A composite number $n$ is weak Giuga if and only if $$ \sum_{p\mid n} \frac{1}{p} - \frac{1}{n} \in\mathbb{N}. $$
AgohGiuga.squarefree_of_isCarmichael A composite Carmichael number is squarefree
AgohGiuga.korselts_criterion A composite number a is Carmichael if and only if it is squarefree and, for all prime p dividing a, we have p - 1 ∣ a - 1
AgohGiuga.isStrongGiuga_iff Giuga showed that a number n is strong Giuga if and only if it is Carmichael and ∑_{p|n} 1/p - 1/n ∈ ℕ (i.e., if and only if it is Carmichael and weak Giuga)
AgohGiuga.agoh_giuga.variants.isStrongGiuga_implies_isCarmichael Every strong Giuga number is a Carmichael number
AgohGiuga.agoh_giuga.variants.le_primeFactors_card_of_isStrongGiuga Giuga showed that a Giuga number has at least 9 prime factors
AgohGiuga.agoh_giuga.variants._1000_le_digits_length_of_isStrongGiuga Giuga showed that a counterexample Giuga number has at least 1000 digits
AgohGiuga.agoh_giuga.variants._1700_le_digits_length_of_isStrongGiuga Bedocchi showed that any Giuga number has at least 1700 digits
AgohGiuga.agoh_giuga.variants._13000_le_digits_length_of_isStrongGiuga Borwein, Borwein, Borwein and Girgensohn showed that any strong Giuga number has at least 13000 digits
AgohGiuga.agoh_giuga.variants.isStrongGiuga_growth Let G(X) denote the number of exceptions n ≤ X to Giuga’s conjecture
AgrawalConjecture.agrawal_conjecture Agrawal's Primality Conjecture
AgrawalConjecture.agrawal_conjecture.variants.popovych Roman B
AlgebraicNormality.IsIrrationalAlgebraic A real number is irrational algebraic if it is algebraic over ℚ but not rational
AlgebraicNormality.isNormalInBase_iff_isEquidistributedModuloOne A real number $x$ is normal in base $b$ if and only if $(b^n x)_{n \ge 0}$ is uniformly distributed modulo $1$ [Wal49], [Bug12, Theorem 4.14]
AlgebraicNormality.irrational_algebraic_absolutely_normal Borel's normality conjecture [Bor50]: every irrational algebraic real is absolutely normal
AlgebraicNormality.irrational_algebraic_normal_in_some_base The weaker normality conjecture: every irrational algebraic real is normal in at least one integer base b ≥ 2
AlgebraicNormality.normal_in_some_base_of_absolutely_normal Absolute normality implies normality in at least one base
AlmostPerfectNumbers.AlmostPerfect A number is almost perfect if the sum of its divisors is equal to $2n - 1$
AlmostPerfectNumbers.exists_almost_perfect_not_power_of_two Non-Power-of-2 Almost Perfect Numbers Conjecture
AmicableNumbers.amicable_220_284 The classic amicable pair $(220, 284)$
AmicableNumbers.IsAmicable.symm IsAmicable is symmetric
AmicableNumbers.relatively_prime_amicable Relatively prime amicable numbers conjecture
AmicableNumbers.infinitely_many_amicable Infinitely many amicable numbers conjecture
AmicableNumbers.opposite_parity_amicable Amicable numbers with opposite parity conjecture
AndrewsCurtis.RelatorTuple An n-tuple of relators in the free group on n generators
AndrewsCurtis.standardRelators The standard relator tuple, consisting of the free generators themselves
AndrewsCurtis.Move The elementary Andrews-Curtis moves: inversion or multiplication of relators, interchange of two relators, and conjugation of one relator by an arbitrary word
AndrewsCurtis.Equivalent Andrews-Curtis equivalence is the equivalence relation generated by elementary moves
AndrewsCurtis.NormallyGenerates A relator tuple normally generates the free group when the normal closure of its range is the whole free group
AndrewsCurtis.andrews_curtis_conjecture The Andrews-Curtis conjecture
Andrica.andrica_conjecture Andrica's conjecture The inequality $\sqrt{p_{n+1}}-\sqrt{p_n} < 1$ holds for all $n$, where $p_n$ is the $n$-th prime number
Andrica.andrica_conjecture.ferreira_large_n Ferreira proved that Andrica's conjecture is true for sufficiently large n
ArtinPrimitiveRootsConjecture.IsDedekindZeta Documented declaration: IsDedekindZeta
ArtinPrimitiveRootsConjecture.DedekindZetaRH The Riemann hypothesis for the Dedekind zeta function
ArtinPrimitiveRootsConjecture.HooleyHypothesis Documented declaration: HooleyHypothesis
ArtinPrimitiveRootsConjecture.S Let $S(a)$ be the set of primes such that $a$ is a primitive root modulo $p$
ArtinPrimitiveRootsConjecture.ArtinConstant Artin's Constant is defined to be the product $$\prod_{p\ \text{prime}} \left(1 - \frac{1}{p(p - 1)}\right)$$
ArtinPrimitiveRootsConjecture.powCorrectionFactor Documented declaration: powCorrectionFactor
ArtinPrimitiveRootsConjecture.entanglementFactor Documented declaration: entanglementFactor
ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.i Artin's Conjecture on Primitive Roots, first half
ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.i Artin's Conjecture on Primitive Roots, first half, conditional on the Riemann hypothesis for the Dedekind zeta functions of the fields
ArtinPrimitiveRootsConjecture.artin_primitive_roots.parts.ii Artin's Conjecture on Primitive Roots, second half
ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.parts.ii Artin's Conjecture on Primitive Roots, second half, conditional on the Riemann hypothesis for the Dedekind zeta functions of the fields
ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_square_or_minus_one Documented declaration: part_ii_square_or_minus_one
ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one Documented declaration: part_ii_power_squarefreePart_not_modeq_one
ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_not_modeq_one Artin's Conjecture on Primitive Roots, second half, power version, conditional on the Riemann hypothesis for the Dedekind zeta functions of the fields
ArtinPrimitiveRootsConjecture.artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one Documented declaration: part_ii_power_squarefreePart_modeq_one
ArtinPrimitiveRootsConjecture.conditional_artin_primitive_roots.variants.part_ii_power_squarefreePart_modeq_one Artin's Conjecture on Primitive Roots, second half, power version, conditional on the Riemann hypothesis for the Dedekind zeta functions of the fields
BalancedPrimes.balanced_primes Let $p_k$ be the $k$-th prime number
BalancedPrimes.balanced_primes_order Let $p_k$ be the $k$-th prime number
BatemanHornConjecture.OmegaP OmegaP S p counts the number of residue classes mod p where at least one polynomial in S vanishes
BatemanHornConjecture.DegreesProduct The product of degrees of polynomials in a finite set
BatemanHornConjecture.BatemanHornPartialProduct The partial Euler product of the Bateman-Horn constant of a set of polynomials S, taken over the primes
BatemanHornConjecture.CountSimultaneousPrimes CountSimultaneousPrimes S x counts the number of n ≤ x at which all polynomials in S attain a prime value
BatemanHornConjecture.bateman_horn_conjecture The Bateman-Horn Conjecture Given a finite collection of distinct irreducible polynomials non-constant
BealConjecture.beal_conjecture The Beal Conjecture: if we are given positive integers $A, B, C, x, y, z$ such that $x, y, z > 2$ and $A^x + B^y = C^z$ then $A, B, C$ have a common divisor
BealConjecture.flt_of_beal_conjecture The Beal Conjecture implies Fermat's last theorem
BeckFiala.beck_fiala_conjecture The Beck–Fiala conjecture There exists a universal constant
BeckFiala.beck_fiala_conjecture.variants.no_sets Sanity check: a system with no sets at all ($m = 0$) admits any colouring vacuously; in particular the all-ones colouring satisfies every discrepancy bound
BetrothedNumbers.IsBetrothed Two natural numbers $m$ and $n$ are betrothed (or quasi-amicable) if $\sigma(m) = \sigma(n) = m + n + 1$, where $\sigma$ is the sum-of-divisors function
BetrothedNumbers.betrothed_48_75 The smallest known betrothed pair $(48, 75)$
BetrothedNumbers.IsBetrothed.symm IsBetrothed is symmetric
BetrothedNumbers.same_parity_betrothed Same parity betrothed numbers conjecture
BetrothedNumbers.infinitely_many_betrothed Infinitude of betrothed numbers conjecture
BingBorsuk.bing_borsuk_conjecture The Bing-Borsuk Conjecture: every $n$-dimensional homogeneous absolute neighborhood retract is a topological $n$-manifold
Bloch.blochRadius The Bloch radius
Bloch.landauRadius The Landau radius $L_f$ of a function $f$ is the supremum of radii of disks contained in the image of the unit disk under $f$
Bloch.blochConstant The Bloch constant $B$ is the largest radius such that every holomorphic function on the unit disk with $f'(0) = 1$ has a schlicht (univalent) disk of that radius in its image
Bloch.blochConstant_lower_bound It is proved in [CP96] that the Bloch constant is bounded below by $\sqrt{3}/4 + 2 \times 10^{-4}$
Bloch.blochConstant_upper_bound It is proved in [AG37] that the Bloch constant is bounded above by $\frac{1}{\sqrt{1 + \sqrt{3}}}\frac{\Gamma(1/3) \Gamma(11/12)}{\Gamma(1/4)}$
Bloch.blochConstant_exact_value Ahlfors and Grunsky also conjectured in [AG37] that this upper bound is the precise value of the Bloch constant
Bloch.univalentBlochConstant The Univalent Bloch constant
Bloch.univalentBlochConstant_lower_bound It is proved in [Skin2009] that the Univalent Bloch constant is bounded below by $0.5708858$
Bloch.univalentBlochConstant_upper_bound The Univalent Bloch constant is trivially bounded above by the Bloch radius of the identity function, which is $1$
Bloch.landauConstant The Landau constant $L$ is the largest radius such that every holomorphic function on the unit disk with $f'(0) = 1$ has a disk of that radius contained in its image
Bloch.landauConstant_lower_bound It is proved in [Ya95] that the Landau constant is bounded below by $0.5 + 10 ^ {-335}$
Bloch.landauConstant_upper_bound It is proved in [Ra43] that the Landau constant is bounded above by $\frac{1}{\sqrt{1 + \sqrt{3}}}\frac{\Gamma(1/3) \Gamma(5/6)}{\Gamma(1/6)}$
Bloch.landauConstant_exact_value In [Ra43], Rademacher says that he strongly believed that this upper bound is the precise value of the Landau constant
Borsuk.HasBorsukCover HasBorsukCover k s means that the set s can be covered by k sets, each of strictly smaller extended diameter than s
Borsuk.BorsukConjecture Borsuk's conjecture in dimension n: every bounded subset of $\mathbb{R}^n$ with at least two points can be partitioned into $n + 1$ sets of strictly smaller diameter
Borsuk.borsuk_conjecture Documented declaration: borsuk_conjecture
Borsuk.borsuk_conjecture.four Borsuk's conjecture in dimension $4$, the smallest open case
Borsuk.borsuk_conjecture.four.nine_parts Partial result in dimension $4$: every bounded subset of $\mathbb{R}^4$ with at least two points can be partitioned into $9$ sets of strictly smaller diameter
Borsuk.borsuk_conjecture.four.eight_parts Partial result in dimension $4$: every bounded subset of $\mathbb{R}^4$ with at least two points can be partitioned into $8$ sets of strictly smaller diameter
BoundedBurnsideProblem.bounded_burnside_problem Let $G$ be a finitely generated group, and assume there exists $n$ such that for every $g$ in $G$, $g^n = 1$
BrennanConjecture.IsUnivalentNormalized The standard class $\mathcal{S}$ of normalised univalent functions on $\mathbb{D}$
BrennanConjecture.integralMeansSpectrum $\beta_f(\tau) := \limsup_{r \to 1^-} \frac{\log \int_{-\pi}^{\pi} |f'(re^{i\theta})|^\tau \, d\theta}{|\log(1-r)|}$
BrennanConjecture.brennan_universalSpectrum Brennan's conjecture, part 1: $B(-2) = 1$
BrennanConjecture.brennan_universalSpectrumBounded Brennan's conjecture, part 2: $B_b(-2) = 1$
BrennanConjecture.brennan_spectra_eq Brennan's conjecture, part 3: $B(-2) = B_b(-2)$
BrennanConjecture.brennan Brennan's conjecture: $B(-2) = B_b(-2) = 1$
Brocard.brocard_conjecture Brocard's Conjecture For every n ≥ 2, between the squares of the n-th and (n+1)-th primes, there are at least four prime numbers
Brocard.brocard_conjecture.ferreira_large_n Ferreira proved that Brocard's conjecture is true for sufficiently large n
Buchi.IsBuchi IsBuchi M asserts that whenever M consecutive values (x + n)² + a (for n = 0, …, M - 1) are all perfect squares, then a must be 0
Buchi.buchi_problem Büchi's problem There exists a positive integer $M$ such that, for all integers $x$ and $a$, if $(x+n)^2 + a$ is a square for $M$ consecutive values of $n$, then $a = 0$
Buchi.buchi_problem_M5 Büchi's problem (first open case, $M = 5$): For all integers $x$ and $a$, if $(x+n)^2 + a$ is a perfect square for $n = 0, 1, 2, 3, 4$, then $a = 0$
Buchi.buchi_false_M0 The case $M = 0$ fails: the hypothesis is vacuous, so $a = 0$ cannot be concluded
Buchi.buchi_false_M1 The case $M = 1$ fails: $0^2 + 4 = 4 = 2^2$ is a square but $a = 4 \neq 0$
Buchi.buchi_false_M2 The case $M = 2$ fails: $(7+n)^2 - 48$ is a perfect square for $n = 0, 1$ ($1^2$ and $4^2$), but $a = -48 \neq 0$
Buchi.buchi_false_M3 The case $M = 3$ fails: $(24+n)^2 - 576$ is a perfect square for $n = 0, 1, 2$ ($0^2$, $7^2$, and $10^2$), but $a = -576 \neq 0$
Buchi.buchi_false_M4 The case $M = 4$ fails: $(246+n)^2 - 60480$ is a perfect square for $n = 0, 1, 2, 3$ ($6^2$, $23^2$, $32^2$, and $39^2$), but $a = -60480 \neq 0$
Bunyakovsky.bunyakovsky_conjecture Bunyakovsky conjecture If a polynomial
BusyBeaver.BB BB(n) is the n-th Busy Beaver number
BusyBeaver.sanity_check To compute BB n, we need only consider machines with states and symbols indexed in Fin
BusyBeaver.BB_1 The value of the Busy Beaver function for 1 state is 1
BusyBeaver.BB_2 The value of the Busy Beaver function for 2 states is 6
BusyBeaver.BB_3 The value of the Busy Beaver function for 3 states is 21
BusyBeaver.BB_4 The value of the Busy Beaver function for 4 states is 107
BusyBeaver.BB_5 The value of the Busy Beaver function for 5 states is 47176870
BusyBeaver.BB_6 Determine the value of the Busy Beaver function at n = 6
CarmichaelTotient.CarmichaelTotientFor Natural number $n$ for which there exists a $m ≠ n$ with $φ(m) = φ(n)$
CarmichaelTotient.carchimichealTotientFor_zero $n = 0 ↔ φ(n) = 0$
CarmichaelTotient.carmichealTotientFor_odd For every odd number $n$, $φ(2n) = φ(n)$
CarmichaelTotient.charmichaelTotient Carmichael's totient function conjecture*: For every positive natural number $n$, there exists a natural number $m$ with $m ≠ n$, such that $φ(n) = φ(m)$
CarmichaelTotient.carchimaelTotient_bound In Theorem 6 in [F1998], Kevin Ford proves that the smallest counterexample to Carmichael's totient function conjecture must be $≥ 10 ^ (10 ^ 10)$
Catalan.pillais_conjecture For positive integers a, b, and c, there are only finitely many positive solutions (x, y, m, n) to the equation $ax^n - by^m = c$ where $(m, n) \neq (2, 2)$ and $x, y > 1$
LebesgueNagell.lebesgue_nagell Lebesgue-Nagell Equation Conjecture For any odd prime $p$, the only integer solutions $(x, y)$ to the equation $x^2 - 2 = y^p$ are $(x, y) = (\pm 1, -1)$
LebesgueNagell.lebesgue_nagell_solution_pos_one The pair $(1, -1)$ is a solution to $x^2 - 2 = y^p$ for any odd $p$
LebesgueNagell.lebesgue_nagell_solution_neg_one The pair $(-1, -1)$ is a solution to $x^2 - 2 = y^p$ for any odd $p$
CernyConjecture.cerny_conjecture Černý Conjecture: Every synchronizing DFA with $n$ states admits a synchronizing word of length at most $(n - 1)^2$
CernyConjecture.shitov_upper_bound Documented declaration: shitov_upper_bound
ClassNumberProblem.class_number_problem There are infinitely many real quadratic fields ℚ(√d) with class number one, where d > 1 is a squarefree integer
ClassNumberProblem.class_number_problem.variants.imaginary Documented declaration: imaginary
CollatzConjecture.collatzStep One step of the Collatz map
CollatzConjecture.collatz_conjecture The Collatz conjecture after CollatzConjecture.collatzStep
CongruentNumber.not_congruentNumber_1 1 is not a congruent number, as proved by Fermat via infinite descent
CongruentNumber.congruentNumber_5 The rational right triangle with side lengths $\frac{3}{2}$, $\frac{20}{3}$, and $\frac{41}{6}$ witnesses that $5$ is a congruent number
CongruentNumber.congruentNumber_6 The $3$-$4$-$5$ right triangle witnesses that $6$ is a congruent number
CongruentNumber.congruentNumber_7 The rational right triangle with side lengths $\frac{35}{12}$, $\frac{24}{5}$, and $\frac{337}{60}$ witnesses that $7$ is a congruent number
CongruentNumber.congruentNumber_157_zagier Zagier's rational right triangle witnesses that $157$ is a congruent number
CongruentNumber.Tunnell_odd Tunnell's theorem (necessary condition) for odd squarefree congruent numbers
CongruentNumber.Tunnell_even Tunnell's theorem (necessary condition) for even squarefree congruent numbers
CongruentNumber.Tunnell_odd_converse Tunnell's theorem (sufficient condition assuming BSD) for odd squarefree congruent numbers
CongruentNumber.Tunnell_even_converse Tunnell's theorem (sufficient condition assuming BSD) for even squarefree congruent numbers
Conway99Graph.completeGraphIsClique A finset of vertices in a complete graph is always a clique
Conway99Graph.completeGraph_cliqueSet The only clique of size n in a complete graph on n vertices is the entire set of vertices
Conway99Graph.NonEdgesAreDiagonals Each two non-adjacent vertices have exactly two common neighbors
Conway99Graph.conway99Graph Documented declaration: conway99Graph
Conway99Graph.triangle_locallyLinear_and_nonEdgesAreDiagonals The triangle is an example with 3 vertices satisfying the condition
Conway99Graph.Conway9 The box product of two triangles is an example with 9 vertices satisfying the condition
DedekindNumber.M $M(n)$ is the number of monotone Boolean functions on $n$ variables
DedekindNumber.IsSperner A Sperner family (antichain) of subsets of Fin n: a family of sets such that no member is a subset of another
DedekindNumber.M' $M'(n)$ is the number of antichains (Sperner families) of subsets of Fin n
DedekindNumber.M_zero Values for small n
DedekindNumber.χ The indicator function of a finset: χ s i = true ↔ i ∈ s
DedekindNumber.supp The support of a Boolean-valued function: supp v = {i | v i = true}
DedekindNumber.toSperner Forward map: monotone function → Sperner family (the minimal true sets)
DedekindNumber.fromSperner Backward map: Sperner family → monotone Boolean function
DedekindNumber.exists_minimal_true_subset Every true set of a monotone Boolean function contains a minimal true set
DedekindNumber.fromSperner_toSperner Converting a monotone function to a Sperner family and back yields the same function
DedekindNumber.toSperner_fromSperner Converting a Sperner family to a monotone function and back yields the same family
DedekindNumber.equivMonotoneSperner The set of monotone Boolean functions on n variables is in bijection with the set of Sperner families of subsets of Fin n
DedekindNumber.kisielewiczFormula A closed formula for the Dedekind numbers as found by Kisielewicz (1998)
DedekindNumber.M_eq_kisielewiczFormula Kisielewicz (1988) proved the following arithmetic formula for the Dedekind numbers
DedekindNumber.M_eq No closed-form expression that allows efficient computation of Dedekind numbers is currently known
DedekindNumber.Dedekind_10 In particular, the Dedekind number for n = 10 is currently unknown
DeterminantalConjecture.determinantal_conjecture Does the determinant of the sum
BabaiSeressConjectures.cayleyGraph The (undirected) Cayley graph of a group $G$ with respect to a generating set $S$
BabaiSeressConjectures.groupDiam The diameter of a finite group $G$, defined as the maximum diameter of the Cayley graphs $\Gamma(G, A)$ over all generating sets $A$ of $G$
BabaiSeressConjectures.groupDiam_fin_one For the trivial group (with one element), the group diameter is zero, since every Cayley graph has only one vertex and hence diameter zero
BabaiSeressConjectures.groupDiam_alternating_three The alternating group
BabaiSeressConjectures.groupDiam_perm_two The symmetric group
BabaiSeressConjectures.babai_seress_conjecture_alternating Babai–Seress Conjecture (Conjecture 1.5): There exists an absolute constant
BabaiSeressConjectures.babai_seress_conjecture Babai–Seress Conjecture (Conjecture 1.7): There exists an absolute constant
Dickson.dickson_conjecture Dickson's conjecture If a finite set of linear integer forms
Dickson.generalized_twin_primes The generalized twin-prime conjecture For any positive integer $k$ there are infinitely many primes $p$ such that $p + 2k$ is prime
Dickson.infinite_safe_primes The infinitude of Sophie Germain primes There are infinitely many primes $p$ such that $2p + 1$ is prime
Dickson.infinite_cousin_primes The infinitude of cousin primes There are infinitely many primes $p$ such that $p + 4$ is prime
Dickson.infinite_sexy_primes The infinitude of sexy primes There are infinitely many primes $p$ such that $p + 6$ is prime
DiophantineTuple.IsDiophantineTuple A finite set s is a Diophantine tuple if each element is nonzero and the product of any two distinct elements is one less than a perfect square
DiophantineTuple.IsIntegralDiophantineTuple An integral Diophantine tuple is a Diophantine tuple of positive integers
DiophantineTuple.IsRationalDiophantineTuple A rational Diophantine tuple is a Diophantine tuple of nonzero rationals
DiophantineTuple.isDiophantineTuple_of_subset The property of being a Diophantine tuple is closed under subsets
DiophantineTuple.fermat_4_tuple There exists a Diophantine 4-tuple; this example is due to Fermat
DiophantineTuple.integralDiophantine4Tuple The set of integral Diophantine 4-tuples
DiophantineTuple.NoIntegralDiophantineFiveTuple The statement that there is no integral Diophantine 5-tuple
DiophantineTuple.noIntegralDiophantineFiveTuple The Diophantine 5-tuple theorem: there does not exist an integral Diophantine 5-tuple
DiophantineTuple.regularExtension Given an integral Diophantine 3-tuple, there is a standard way to extend it to a 4-tuple by adjoining the value of this function
DiophantineTuple.HasUniqueExtension The property that the Diophantine 3-tuple $\{a, b, c\}$ extends to a 4-tuple $\{a, b, c, d\}$ with $d > \max(a, b, c)$ only via regularExtension
DiophantineTuple.HasUniqueExtensionOfForall The statement that every integral Diophantine triple of three distinct elements has a unique extension by a larger element
DiophantineTuple.hasUniqueExtension_of_forall The "strong Diophantine 5-tuple conjecture", so-called because it implies the Diophantine 5-tuple theorem (see noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall)
DiophantineTuple.noIntegralDiophantineFiveTuple_of_hasUniqueExtensionOfForall If every integral Diophantine triple has a unique extension by a larger element, then there is no integral Diophantine 5-tuple
DiophantineTuple.hasUniqueExtension_of_1_3_8 HasUniqueExtension is known to hold for certain triples, including
DiophantineTuple.gibbs_6_tuple A known rational Diophantine 6-tuple
DiophantineTuple.rational_7_tuple Does there exist a rational Diophantine 7-tuple?
ElliottHalberstamConjecture.primesInAPCount $\pi(x; q, a)$: the number of primes $p \le x$ with $p \equiv a \pmod q$
ElliottHalberstamConjecture.E The error term
ElliottHalberstamConjecture.elliott_halberstam Documented declaration: elliott_halberstam
ElliottHalberstamConjecture.elliott_halberstam.variants.bombieri_vinogradov The Bombieri–Vinogradov theorem: the Elliott–Halberstam conjecture holds for every $\theta < 1/2$
ElliottHalberstamConjecture.elliott_halberstam.variants.friedlander_granville The Elliott–Halberstam conjecture fails at the endpoint $\theta = 1$, as shown by Friedlander and Granville [FG89]
EllipticCurveRank.RatEllipticCurve A data structure representing isomoprhism classes of elliptic curves over ℚ
EllipticCurveRank.RatEllipticCurve.toWeierstrass Convert the structure RatEllipticCurve to a Weierstrass curve
EllipticCurveRank.RatEllipticCurve.rank The rank of an elliptic curve over ℚ
EllipticCurveRank.RatEllipticCurve.naiveHeight The naïve height of an elliptic curve over ℚ
EllipticCurveRank.RatEllipticCurve.heightLE The set of elliptic curves over ℚ with naïve height less than or equal to a given height
EllipticCurveRank.RatEllipticCurve.card_heightLE_div_pow_five_div_six_tensto Formula (5.1.1) of [PPVW2016]: The number of elliptic curves over ℚ with naïve height at most H is asymptotically 2^(4/3)*3^(-3/2)/ζ(10) * H^(5/6)
EllipticCurveRank.RatEllipticCurve.half_rank_zero_and_half_rank_one Conjecture by Goldfeld and Katz–Sarnak: if elliptic curves over ℚ are ordered by their heights, then 50% of the curves have rank 0 and 50% have rank 1
EllipticCurveRank.RatEllipticCurve.avg_rank_lt_0885 Theorem 3 of [BS2013]: when elliptic curves over ℚ are ordered by height, their average rank is < .885
EllipticCurveRank.RatEllipticCurve._08375_le_density_rank_zero_one Theorem 4 of [BS2013]: when elliptic curves over ℚ are ordered by height, a density of at least 83.75% have rank 0 or 1
EllipticCurveRank.RatEllipticCurve._02062_le_density_rank_zero Theorem 5 of [BS2013]: when elliptic curves over ℚ are ordered by height, a density of at least 20.62% have rank 0
EllipticCurveRank.RatEllipticCurve.quadraticTwist The quadratic twist $E^d$ of $E : y^2 = x^3 + Ax + B$ by an integer $d$, given by the Weierstrass equation $y^2 = x^3 + d^2Ax + d^3B$
EllipticCurveRank.RatEllipticCurve.quadraticTwist_one The twist of $E$ by $d = 1$ is $E$ itself
EllipticCurveRank.RatEllipticCurve.isElliptic_quadraticTwist The quadratic twist of an elliptic curve over $\mathbb{Q}$ by a nonzero $d$ is again an elliptic curve
EllipticCurveRank.RatEllipticCurve.naiveHeight_quadraticTwist The naïve height of the Weierstrass model $E^d$ is $|d|^6$ times the naïve height of $E$
EllipticCurveRank.RatEllipticCurve.twistRank The rank of the quadratic twist $E^d$ of an elliptic curve $E$ over $\mathbb{Q}$
EllipticCurveRank.RatEllipticCurve.twistIndexLE The set of nonzero integers $d$ with $|d| \leq H$, which index the quadratic twists $E^d$ ordered by $|d|$
EllipticCurveRank.RatEllipticCurve.ncard_twistIndexLE There are $2H$ nonzero integers $d$ with $|d| \leq H$
EllipticCurveRank.RatEllipticCurve.goldfeld_conjecture Goldfeld's conjecture ([Goldfeld1979], Conjecture B), in the form stated in the introduction of [Smith2025]: when the quadratic twists
EllipticCurveRank.RatEllipticCurve.goldfeld_conjecture.variants.two_le Goldfeld's conjecture ([Goldfeld1979], Conjecture B), in the form stated in the introduction of [Smith2025], in the case
EllipticCurveRank.RatEllipticCurve.goldfeld_conjecture.variants.squarefree Documented declaration: squarefree
EllipticCurveRank.RatEllipticCurve.unbounded_rank_conjecture From [PPVW2016], Section 3.1: "from the mid-1960s to the present, it seems that most experts conjectured unboundedness."
EllipticCurveRank.RatEllipticCurve.finite_twentyone_lt_finrank From [PPVW2016], Section 8.2: "Our heuristic predicts (a) All but finitely many E ∈ ℰ satisfy rk E(ℚ) ≤ 21"
EllipticCurveRank.RatEllipticCurve.rank_height_count_asymptotic [PPVW2016] 8.2(b): for 1 ≤ r ≤ 20, the number of elliptic curves over ℚ with rank at least r and naïve height at most H is asymptotically H ^ ((21 - r) / 24 + o(1))
EllipticCurveRank.RatEllipticCurve.twentyone_le_rank_height_count_asymptotic [PPVW2016] 8.2(c): the number of elliptic curves over ℚ with rank ≥ 21 and naïve height at most H is asymptotically at most H ^ o(1)
EllipticCurveRank.RatEllipticCurve.exists_rank_ge_thirtyone Is there an elliptic curve over ℚ of rank at least 31?
EllipticCurveRank.RatEllipticCurve.exists_rank_ge_thirtytwo Is there an elliptic curve over ℚ of rank at least 32?
EllipticCurveRank.WeierstrassCurve.claudeAlpogeHowell31 The elliptic curve over ℚ of rank at least 31 found by Claude, Levent Alpöge and Ava Howell in 2026
EllipticCurveRank.WeierstrassCurve.Δ_claudeAlpogeHowell31 See https://elliptic-rank.icarm.cloud/curve/302
EllipticCurveRank.WeierstrassCurve.thirtyone_le_rank_claudeAlpogeHowell31 The rank of the Claude–Alpöge–Howell curve is at least 31
EllipticCurveRank.WeierstrassCurve.rank_claudeAlpogeHowell31 The rank of the Claude–Alpöge–Howell curve is exactly 31
EllipticCurveRank.WeierstrassCurve.ranksunbounded30 The elliptic curve over ℚ of rank at least 30 found by user ranksunbounded in 2026
EllipticCurveRank.WeierstrassCurve.Δ_ranksunbounded30 See https://elliptic-rank.icarm.cloud/curve/273
EllipticCurveRank.WeierstrassCurve.thirty_le_rank_ranksunbounded30 The rank of the ranksunbounded curve is at least 30
EllipticCurveRank.WeierstrassCurve.rank_ranksunbounded30 The rank of the ranksunbounded curve is exactly 30
EllipticCurveRank.WeierstrassCurve.elkiesKlagsbrun29 The elliptic curve over ℚ of rank at least 29 found by Elkies and Klagsbrun in 2024
EllipticCurveRank.WeierstrassCurve.Δ_elkiesKlagsbrun29 See https://mathoverflow.net/a/478050
EllipticCurveRank.WeierstrassCurve.twentynine_le_rank_elkiesKlagsbrun29 The rank of the Elkies-Klagsbrun curve is at least 29
EllipticCurveRank.WeierstrassCurve.rank_elkiesKlagsbrun29 The rank of the Elkies-Klagsbrun curve is exactly 29
EllipticCurveRank.WeierstrassCurve.elkies28 The elliptic curve over ℚ of rank at least 28 found by Elkies in 2006
EllipticCurveRank.WeierstrassCurve.Δ_elkies28 See https://mathoverflow.net/a/478050
EllipticCurveRank.WeierstrassCurve.twentyeight_le_rank_elkies28 The rank of the Elkies curve is at least 28
EllipticCurveRank.WeierstrassCurve.rank_elkies28 The rank of the Elkies curve is exactly 28
ErdosMoser.powerSum The power sum $S_k(m)=\sum_{i=1}^{m-1} i^k$
ErdosMoser.erdos_moser_conjecture The only positive solution of $S_k(m)=m^k$ is $(k,m)=(1,3)$
ErdosMoser.powerSum_one_three The pair $(1,3)$ is the trivial solution of the Erdős–Moser equation
ErdosMoser.powerSum_two_three For $k=2$ and $m=3$, the left side is $1^2+2^2=5$, not $3^2$
ErdosMoser.powerSum_zero Zero is a solution for every positive exponent, so the conjecture must require $m>0$
EuclidNumbers.Euclid The n-th Euclid number is the product of the first n prime numbers plus one
EuclidNumbers.infinite_prime_euclid_numbers It is not known whether there is an infinite number of prime Euclid numbers
EuclidNumbers.euclid_numbers_are_square_free It is not known whether every Euclid number is a square-free number
EulerBrick.IsEulerBrick An Euler brick is a rectangular cuboid where all edges and face diagonals have integer lengths
EulerBrick.IsPerfectCuboid A perfect cuboid is an Euler brick with an integer space diagonal
EulerBrick.IsEulerHyperBrick Generalization of an Euler brick to $n$-dimensional space
EulerBrick.perfect_euler_brick_existence Is there a perfect Euler brick?
EulerBrick.four_dim_euler_brick_existence Is there an Euler brick in $4$-dimensional space?
EulerBrick.n_dim_euler_brick_existence Is there an Euler brick in $n$-dimensional space for any $n > 3$?
EulerBrick.CuboidOneFor Pairs of natural numbers for which the first Cuboid polynomial is irreducible
EulerBrick.CuboidOne First Cuboid conjecture*: For all positive coprime integers $a$, $b$ with $a ≠ b$, the polynomial of the first Cuboid polynomial is irreducible
EulerBrick.CuboidTwoFor Pairs of natural numbers for which the second Cuboid polynomial is irreducible
EulerBrick.CuboidTwo Second Cuboid conjecture*: For all positive coprime integers $a$, $b$ with $a ≠ b$, the polynomial of the second Cuboid polynomial is irreducible
EulerBrick.cuboidTwo The second Cuboid conjecture
EulerBrick.CuboidThreeFor Triplets of natural numbers for which the third Cuboid polynomial is irreducible
EulerBrick.CuboidThree Third Cuboid conjecture*: For all positive, pairwise different coprime integers
EulerBrick.cuboidThree The third Cuboid conjecture
EulerBrick.cuboid_perfect_euler_brick In [Sh12], Ruslan notes that a perfect Euler brick does not exist if all three Cuboid conjectures hold
EulerSumOfPowers.eulers_sum_of_powers_conjecture Euler's sum of powers conjecture states that for integers
EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k4 Euler's sum of powers conjecture is false for $k=4$ (counterexample exists)
EulerSumOfPowers.eulers_sum_of_powers_conjecture.false_for_k5 Euler's sum of powers conjecture is false for $k=5$ (counterexample exists)
Exponentials.four_exponentials_conjecture Four exponentials conjecture Let $x_0, x_1$ and $y_0, y_1$ be $\mathbb Q$-linearly independent pairs of complex numbers, then some $e^{x_i y_j}$ is transcendental
Exponentials.two_pow_three_pow_transcendental The four exponential conjecture would imply that for any irrational number $t$, at least one of the numbers $2^t$ and $3^t$ is transcendental
FactorialPrime.IsFactorialPrime A factorial prime is a prime one above or one below a factorial
FactorialPrime.infinitely_many_factorial_primes There are infinitely many factorial primes
Falconer.falconer_conjecture_two Falconer's distance set conjecture, $d = 2$ case
Falconer.falconer_conjecture Falconer's distance set conjecture in dimension $d \ge 2$
FeitThompsonPrimeConjecture.feit_thompson_primes There are no distinct primes $p$ and $q$ such that $\frac{q^p - 1}{q - 1}$ divides $\frac{p^q - 1}{p - 1}$
Fermat.fermat_number_are_composite Are Fermat numbers composite for all n > 4?
Fermat.infinite_fermat_primes Are there infinitely many Fermat primes?
Fermat.infinite_fermat_composite Are there infinitely many composite Fermat numbers?
Fermat.exists_fermat_not_squarefree Does a Fermat number exist that is not square-free?
FermatCatalanConjecture.FermatCatalanSet' The set of solutions to the Fermat-Catalan Conjecture, i.e. the set of solutions $(a,b,c,m,n,k)$ to the equation $a^m + b^n = c^k$ where $\frac 1 m + \frac 1 n + \frac 1 k < 1$
FermatCatalanConjecture.fermatCatalanConjecture The proposition that the Fermat-Catalan Conjecture is true
FermatCatalanConjecture.fermat_catalan The Fermat–Catalan conjecture states that the equation
FermatCatalanConjecture.fermat_catalan.variants.darmon_granville By the Darmon-Granville theorem, for any fixed choice of positive integers m, n and k satisfying
FibonacciPrimes.fib_primes_infinite Documented declaration: fib_primes_infinite
FibonacciPrimes.fib_primes_infinite.variant There are infinitely many indices $i$, such that the $i$-th Fibonacci is prime
FibonacciPrimes.indices_infinite_iff_fib_primes_infinite The two ways of phrasing the conjecture are equivalent
Firoozbakht.firoozbakhtSeq The sequence of $\sqrt[n]{p_n}$ where $p_n$ is the n:th prime number
Firoozbakht.firoozbakht_conjecture Firoozbakht's conjecture The inequality $\sqrt[n+1]{p_{n+1}} < \sqrt[n]{p_n}$ holds for all prime numbers $p_n$
Firoozbakht.firoozbakht_conjecture_consequence The inequality $p_{n+1}-p_n < (\log p_n)^2-\log p_n$ holds for all $n>4$
FlintCooksonHills.flint_hills_series_converges The Flint Hills series summing $csc(n)^2 / n^3$ from $n=1$ to $\infty$ converges
FlintCooksonHills.cookson_hills_series_converges The Cookson Hills series summing $sec(n)^2 / n^3$ from $n=1$ to $\infty$ converges
FortuneConjecture.exists_one_lt_prime_add For any natural number N there is some m > 1 with N + m prime; an immediate consequence of the infinitude of primes
FortuneConjecture.fortunateNumber The $n$-th Fortunate number (0-indexed): the smallest integer $m > 1$ such that $p_{n+1}\\# + m$ is prime
FortuneConjecture.fortunateNumber_spec fortunateNumber n is greater than $1$, and adding it to the primorial of the $(n+1)$-st prime yields a prime
FortuneConjecture.fortunateNumber_le Minimality of fortunateNumber n: no smaller integer $m > 1$ makes primorial (Nat.nth Nat.Prime n) + m prime
FortuneConjecture.fortune_conjecture Fortune's Conjecture: Every Fortunate number is prime
Fuglede.FugledeConjectureFor Fuglede's conjecture in dimension n: A bounded subset of ℝ^n with positive Lebesgue measure is spectral iff it tiles ℝ^n by translation
Fuglede.FugledeConjecture.variants.dim_1 Fuglede's conjecture in one dimension: A bounded subset of ℝ with positive Lebesgue measure is spectral iff it tiles ℝ by translation
Fuglede.FugledeConjecture.variants.dim_2 Fuglede's conjecture in two dimensions: A bounded subset of ℝ^2 with positive Lebesgue measure is spectral iff it tiles ℝ^2 by translation
Fuglede.FugledeConjecture.variants.dim_3_or_higher Fuglede's conjecture fails in every dimension
GapConjecture.gap_conjecture If a finitely generated group has superpolynomial growth, then with respect to any finite generating set its growth function is at least
GaussCircleProblem.N Let $N(r)$ be the number of points $(m, n)$ within a circle of radius $r$, where $m$ and $n$ are both integers
GaussCircleProblem.E Let $E(r)$ be the error term between the number of integral points inside the circle and the area of the circle; that is $N(r) = \pi r^2 + E(r)$
GaussCircleProblem.error_le Gauss proved that $$ |E(r)|\leq 2\sqrt{2}\pi r, $$ for sufficiently large $r$
GaussCircleProblem.error_not_isLittleO Hardy and Laundau independently found a lower bound by showing that $$ |E(r)| \neq o\left(r^{1/2}(\log r)^{1/4}\right) $$
GaussCircleProblem.error_isBigO It is conjectured that the correct bound is $$ |E(r)| = O\left(r^{1/2 + o(1)}\right) $$ [Ha59] Hardy, G
GaussCircleProblem.exact_form_floor The value of $N(r)$ can be expressed as $$ N(r) = 1 + 4\sum_{i=0}^{\infty}\left(\left\lfloor\frac{r^2}{4i+1}\right\rfloor - \left\lfloor\frac{r^2}{4i + 3}\right\rfloor\right). $$
Gilbreath.d Gilbreath's nth difference, $d^n$ Let $d^0(n) = p_n$ and $d^k(n) = |d^{k-1}(n+1) - d^{k-1}(n)|$
Gilbreath.gilbreath_conjecture Gilbreath's conjecture Gilbreath's conjecture states that every term in the sequence $d^k_0$ for $k > 0$ is equal to 1
GoldbachConjecture.goldbach Goldbach's conjecture
TernaryGoldbachConjecture.ternaryGoldbach Ternary Goldbach
GoodmanConjecture.IsPValent Documented declaration: IsPValent
GoodmanConjecture.isPValent_id_one The identity is $1$-valent (univalent) on the unit disk
GoodmanConjecture.coeff The $n$-th Taylor coefficient of $f$ at $0$, i.e. $b_n = f^{(n)}(0) / n!$
GoodmanConjecture.IsNormalized Goodman's normalisation $f(z) = \sum_{n \ge 1} b_n z^n$, i.e. $f(0) = 0$
GoodmanConjecture.goodmanBound The Goodman bound $$\sum_{k=1}^{p} \frac{2k (n+p)!}{(p-k)!\,(p+k)!\,(n-p-1)!\,(n^2-k^2)} |b_k|,$$ the conjectured upper bound for $|b_n|$
GoodmanConjecture.goodman_conjecture Goodman's conjecture
GoodmanConjecture.goodmanBound_one Sanity check: in the univalent case $p = 1$, the Goodman bound reduces to the classical Bieberbach-type bound $|b_n| \le n\,|b_1|$
Goormaghtigh.repunit The repunit with digits digits in the given base, expressed without division
Goormaghtigh.IsGoormaghtighNumber A number having repunit representations of at least three digits in two distinct bases
Goormaghtigh.goormaghtigh_conjecture The only Goormaghtigh numbers are $31$ and $8191$
Goormaghtigh.isGoormaghtighNumber_31 The number $31$ is a repunit in bases $2$ and $5$
Goormaghtigh.isGoormaghtighNumber_8191 The number $8191$ is a repunit in bases $2$ and $90$
Goormaghtigh.repunit_two_digits Allowing two-digit repunits would make $13$ a representation in two distinct bases
GracefulLabeling.graceful_tree_conjecture Every tree admits a graceful labeling
Grimm.grimm_conjecture Grimm's Conjecture If $n, n+1, \dots, n+k-1$ are all composite numbers, then there are $k$ distinct primes $p_i$ such that $p_i$ divides $n + i$ for all $0 \le i \le k-1$
Grimm.grimm_conjecture_weak Grimm's Conjecture, weaker version If $n, n+1, \dots, n+k-1$ are all composite numbers, then their product has at least $k$ distinct prime divisors
GromovPolynomialGrowth.growthFunction_not_polynomial_of_infinite Documented declaration: growthFunction_not_polynomial_of_infinite
GromovPolynomialGrowth.GromovPolynomialGrowthTheorem Gromov's Polynomial Growth Theorem : A finitely generated group has polynomial growth if and only if it is virtually nilpotent
Hadamard.IsHadamard A square matrix $M$ with $±1$-entries that satisfies the equality $|M| ≤ n^\frac{n}{2}$ is called a Hadamard matrix
Hadamard.IsHadamard' Equivalently, a square matrix $M$ with $±1$-entries $|A| ≤ n^\frac{n}{2}.$ if it satisfies the equality $M^TM = n \cdot 1$, where $1$ denotes the unit matrix
Hadamard.isHadamard_equiv_isHadamard' Both definitions are equivalent
Hadamard.HadamardConjecture There exists a Hadamard matrix for all $n = 4k$
Hadamard.H12 Hadamard constructs a 12 x 12 matrix
Hadamard.isHadamard_H12 which satisfies the condition
Hadamard.HadamardConjecture.variants.first_cases For all $k ≤ 166$, it is known there that there is a Hadamard matrix of size $4 * k$
Hadamard.HadamardConjecture.variants.«167» The smallest order for which no Hadamard matrix is presently known is $668 = 4 * 167$
Hall.hall_conjecture Original Hall's conjecture with exponent $1/2$
Hall.elkies_bound Elkies' example $(x, y) = (5853886516781223, 447884928428402042307918)$ shows that such $C$ must be less than $0.0215$
Hall.danilov Danilov proved that one cannot replace the exponent $1/2$ with larger number
Hall.weak_hall_conjecture Weak form of Hall's conjecture: relax the exponent from $1/2$ to $1/2 - \varepsilon$
HardyLittlewood.IsPrimeConstellation A prime constellation is a tuple $(p, p + m_1, \dots, p + m_k)$ such that the $m_i$ are all positive even integers and every entry is a prime number
HardyLittlewood.IsAdmissiblePrimeConstellation A prime constellation is said to be admissible if its elements do not form a complete set of residue classes with respect to any prime
HardyLittlewood.Nat.numResidues The number of distinct residue classes amongst a tuple $(m_0, \dots, m_k)$ for a prime $q$
HardyLittlewood.Nat.primeTupleCounting For a given tuple $(m_1, \dots, m_k)$, this counts number of admissible prime constellations $(p, p + m_1, \dots, p + m_k)$ where $p \leq n$
HardyLittlewood.FirstHardyLittlewoodConjectureFor The first Hardy–Littlewood conjecture for the tuple
HardyLittlewood.first_hardy_littlewood_conjecture Let $P = (m_1, \dots, m_k)$ be a tuple of distinct positive even integers
HardyLittlewood.SecondHardyLittlewoodConjectureFor Wikipedia URL: https://en.wikipedia.org/wiki/Second_Hardy%E2%80%93Littlewood_conjecture ## Second Hardy-Littlewood Conjecture
HardyLittlewood.second_hardy_littlewood_conjecture For integers $x, y \geq 2$, $$ \pi(x + y) \leq \pi(x) + \pi(y), $$ where $\pi(z)$ denotes the prime-counting function, giving the number of primes up to and including $z$
HardyLittlewood.not_first_and_secondHardyLittlewoodConjecture Richards [Ri74] showed that only one of the two Hardy-Littlewood conjectures can be true
HasseWeil.HasMeromorphicContinuation Documented declaration: HasMeromorphicContinuation
HasseWeil.HasAnalyticContinuation Documented declaration: HasAnalyticContinuation
HasseWeil.HasAnalyticContinuation.hasMeromorphicContinuation An analytic continuation is a meromorphic continuation
HasseWeil.HasMeromorphicContinuation.unique A meromorphic continuation is unique away from isolated points: two meromorphic continuations of the $L$-series of E agree on a punctured neighbourhood of every point
HasseWeil.exists_hasMeromorphicContinuation The $L$-series of an elliptic curve over a number field has a meromorphic continuation to $\mathbb{C}$
HasseWeil.exists_hasAnalyticContinuation The Hasse--Weil conjecture ([Tate1966], p
HasseWeil.exists_hasAnalyticContinuation_rat The Hasse--Weil conjecture over $\mathbb{Q}$: the $L$-series of an elliptic curve over $\mathbb{Q}$ has an analytic continuation to $\mathbb{C}$
Idoneal.IsIdoneal Equivalent definition: A positive integer $n$ is idoneal if and only if it cannot be written as $ab + bc + ac$ for distinct positive integers $a, b,$ and $c$
Idoneal.knownIdonealNumbers The 65 known idoneal numbers that are conjectured to be the only idoneal numbers
Idoneal.exists_triple_iff_bounded Reduces the unbounded search for a representation n = a*b + b*c + a*c (with 0 < a < b < c) to a bounded, decidable double search over a, b ∈ range (n+1)
Idoneal.knownIdonealNumbers_are_idoneal All 65 known idoneal numbers are indeed idoneal
Idoneal.idoneal_numbers_completeness Idoneal numbers completeness conjecture
InscribedSquare.IsRectangle Documented declaration: IsRectangle
InscribedSquare.inscribed_square_problem Inscribed square problem Does every Jordan curve admit an inscribed square?
InscribedSquare.inscribed_rectangle_problem Inscribed rectangle problem Does every Jordan curve admit inscribed rectangles of any given aspect ratio?
InscribedSquare.exists_inscribed_rectangle It is known that every Jordan curve admits at least one inscribed rectangle
InscribedSquare.exists_inscribed_rectangle_of_smooth It is known that every smooth Jordan curve admits inscribed rectangles of all aspect ratios
InscribedSquare.exists_inscribed_square_of_C2 It is also known that every $C^2$ Jordan curve admits an inscribed square
InvariantSubspaceProblem.ClosedInvariantSubspace Documented declaration: ClosedInvariantSubspace
InvariantSubspaceProblem.Invariant_subspace_problem Documented declaration: Invariant_subspace_problem
InvariantSubspaceProblem.Invariant_subspace_problem_finite_dimensional Every (bounded) linear operator T : H → H on a finite-dimensional normed space H of dimension at least 2 has a non-trivial (closed) T-invariant subspace
InvariantSubspaceProblem.Invariant_subspace_problem_non_separable Every bounded linear operator T : H → H on a non-separable Hilbert space H has a non-trivial closed T-invariant subspace
InvariantSubspaceProblem.Invariant_subspace_problem_normal_operator Every normal linear operator T : H → H on a Hilbert space H of dimension at least 2 has a non-trivial closed T-invariant subspace
InvariantSubspaceProblem.Invariant_subspace_problem_l1 Documented declaration: Invariant_subspace_problem_l1
InverseGalois.IsRealizable Say a group G is realizable over a field K if it is isomorphic to the Galois group of a Galois extension of K
InverseGalois.inverse_galois_problem The Inverse Galois Problem: every finite group is isomorphic to the Galois group of a Galois extension of the rationals
InverseGalois.inverse_galois_problem.variants.cyclic Every finite cyclic group is realizable
InverseGalois.inverse_galois_problem.variants.abelian Every finite abelian group is realizable
InverseGalois.inverse_galois_problem.variants.symmetric_group Every finite symmetric group is realizable
InverseGalois.inverse_galois_problem.variants.complex_rational_functions Every finite group is realisable over the field of rational functions with complex coefficients
InverseGalois.inverse_galois_problem.variants.complex_function_field Every finite group is realisable over the field of rational functions with coefficients K, where K is any algebraically closed field of characteristic 0
Irrational.algebraicIndependent_e_pi Are $e$ and $\pi$ algebraically independent?
Irrational.irrational_e_plus_pi Is $e + \pi$ irrational?
Irrational.irrational_e_times_pi Is $e \pi$ irrational?
Irrational.irrational_e_to_e Is $e ^ e$ irrational?
Irrational.irrational_pi_to_e Is $\pi ^ e$ irrational?
Irrational.irrational_pi_to_pi Is $\pi ^ \pi$ irrational?
Irrational.irrational_ln_pi Is $\ln(\pi)$ irrational?
Irrational.irrational_eulerMascheroniConstant Is the Euler-Mascheroni constant $\gamma$ irrational?
Irrational.irrational_catalanConstant Is the Catalan constant $$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$ irrational?
JacobianConjecture.F Alpöge/Fable's counterexample: a polynomial self-map of k³ with Jacobian determinant -2 which is not injective
JacobianConjecture.G A variant of Alpöge/Fable's counterexample: a polynomial self-map of k³ with Jacobian determinant 1 which is not injective
JacobianConjecture.aeval_F_eq F identifies the two distinct points (0, 0, -1/4) and (1, -3/2, 13/2)
JacobianConjecture.aeval_G_eq G identifies the two distinct points (1, 0, 1) and (0, 3, -71)
JacobianConjecture.JacobianConjectureProp The predicate that the Jacobian conjecture holds for a given field and variable index type (i.e. number of variables)
JacobianConjecture.jacobian_conjecture Documented declaration: jacobian_conjecture
JacobianConjecture.jacobian_conjecture_two_variables Does the Jacobian conjecture hold in the two variable case?
JacobianConjecture.sanity_check_condition_1 Let's check that we've stated the "invertible Jacobian" condition correctly by proving an equivalence
JacobianConjecture.jacobian_conjecture_identity Let's apply the conjecture to a trivial case to make sure things are working as expected
Jacobson.JacobsonConjectureFor The Jacobson conjecture for a ring $R$: the infimum of the powers of the Jacobson radical $J$ is the zero ideal
Jacobson.jacobson_conjecture The Jacobson conjecture (in its modern form): In a (noncommutative) ring which is left and right Noetherian, the intersection of the powers of the Jacobson ideal is trivial
Jacobson.jacobson_conjecture_of_comm_ring For commutative rings this is the case as a consequence of Krull's intersection theorem
Jacobson.jacobson_conjecture_of_right_noetherian Originally, on page 200 of [Ja1956], Jacobson asked if the Jacobson conjecture holds for all right Noetherian rings
JugglerConjecture.jugglerStep Consider the following operation on the natural numbers: If the number is even, take the floor of the square root
JugglerConjecture.juggler_conjecture Now form a sequence beginning with any positive integer, where each subsequent term is obtained by applying the operation defined above to the previous term
JugglerConjecture.jugglerStep_36 Example: jugglerStep 36 = ⌊36^(1/2)⌋ = ⌊6⌋ = 6 (since 36 is even)
Kakeya.IsKakeya A set S in ℝⁿ is called a Kakeya set if it contains a unit line segment in every direction
Kakeya.isKakeya_closedBall A trivial example: the closed ball of radius 1 in ℝⁿ is a Kakeya set
Kakeya.KakeyaSetConjectureDim The Kakeya set conjecture in dimension n: the statement that every Kakeya set in ℝⁿ has Hausdorff dimension n
Kakeya.kakeya_set_conjecture The Kakeya set conjecture: Kakeya sets in $\mathbb{R}^n$ have Hausdorff dimension $n$
Kakeya.kakeya_2d The two-dimensional case, proved by Davies [Da71]
Kakeya.kakeya_3d The three-dimensional case, proved by Wang, Zahl [WaZa25]
Kakeya.IsKakeyaFinite A finite field variant of the Kakeya problem considers subsets of 𝔽_qⁿ that contain a line in every direction
Kakeya.kakeya_finite The finite field Kakeya conjecture asserts that any Kakeya set in 𝔽_qⁿ has size at least c_n · qⁿ for some constant c_n depending only on n
Kaplansky.zero_divisor_conjecture The zero-divisor conjecture If G is torsion-free, then the group algebra K[G] has no non-trivial zero divisors
Kaplansky.idempotent_conjecture The idempotent conjecture If G is torsion-free, then K[G] has no non-trivial idempotents
Kaplansky.IsTrivialUnit A unit in K[G] is trivial if it is exactly of the form kg where: - k is a unit in the base field K - g is an element of the group G
Kaplansky.PromislowGroup The Promislow group ⟨ a, b | b⁻¹a²ba², a⁻¹b²ab² ⟩
Kaplansky.promislow_group_is_torsionfree The Promislow group is torsion-free
Kaplansky.UnitConjecture.counterexamples.i If $P$ is the Promislow group, then the group ring $\mathbb{F}_p[P]$ has a non-trivial unit
Kaplansky.UnitConjecture.counterexamples.ii If $P$ is the Promislow group, then the group ring $\mathbb{C}[P]$ has a non-trivial unit
Kaplansky.counter_unit_conjecture The Unit Conjecture is false
Kaplansky.counter_unit_conjecture_weak There is a counterexample to Unit Conjecture in any characteristic
Koethe.IsNil Say a subset I of a ring R is nilpotent if all its elements are nilpotent
Koethe.KotheRadical The Kothe Radical of a ring R is the sum of all (two-sided) nil ideals of R
Koethe.IsNil.mono A subset of a nil subset is nil
Koethe.isNil_bot The zero two-sided ideal is nil
Koethe.IsNil.sup The sum of two nil two-sided ideals is nil
Koethe.isNil_kotheRadical The Köthe radical is a nil ideal: every element of Nil* R is nilpotent
Koethe.IsNil.map_mulRight If L is a nil left ideal, then so is L * y for any y
Koethe.KotheConjecture The Köthe conjecture: In any ring, the sum of two nil left ideals is nil
Koethe.KotherConjecture.variants.le_KotherRadical The Köthe conjecture: every left nil radical is contained in the Köthe radical
Koethe.KotherConjecture.variants.general_matrix The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_n(I) is a nil ideal of the matrix ring M_n(R)
Koethe.KotherConjecture.variants.two_by_two_matrix The Köthe conjecture: for any nil ideal I of R, the matrix ideal M_2(I) is a nil ideal of the matrix ring M_2(R)
Koethe.KotherConjecture.variants.matrixOver_KotherRadical The Köthe conjecture: for any finite index type n, the Köthe radical of the matrix ring M_n(R) is the ideal M_n(Nil*(R)) of matrices with entries in the Köthe radical of R
Koethe.columnIdeal The left ideal of Matrix n n R consisting of the matrices with entries in I which are supported on the j-th column
Koethe.pow_succ_apply_of_mem_columnIdeal Powers of a matrix supported on the j-th column are controlled by the (j, j) entry
Koethe.isNil_columnIdeal For a nil ideal I, the column ideals columnIdeal I j are nil left ideals
Koethe.mem_iSup_columnIdeal The matrix ideal M_n(I) is the sum of the column ideals columnIdeal I j
Koethe.isNil_iSup_of_kotheConjecture If the Köthe conjecture holds (for rings in universe w), then a finite sum of nil left ideals is nil
Koethe.KotherConjecture.variants.general_matrix_of_KotheConjecture The Köthe conjecture (for rings in universe max u v) implies the general matrix formulation for rings in universe u and index types in universe v
Koethe.KotherConjecture.variants.KotheConjecture_of_le_KotherRadical If every nil left ideal is contained in the Köthe radical, then the Köthe conjecture holds
Koethe.KotherConjecture.variants.le_KotherRadical_of_KotheConjecture The Köthe conjecture implies that every nil left ideal is contained in the Köthe radical
Koethe.KotherConjecture.variants.general_matrix_of_le_KotherRadical If every nil left ideal is contained in the Köthe radical (for rings in universe max u v), then the general matrix formulation holds
Koethe.KotherConjecture.variants.two_by_two_matrix_of_general_matrix The general matrix formulation of the Köthe conjecture trivially implies the 2 × 2 one
Koethe.isNil_matrix_of_equiv Nilness of the matrix ideal M_n(I) only depends on the cardinality of n
Koethe.isNil_matrix_of_sum If M_{n ⊕ m}(I) is nil then so is M_n(I), using the block embedding
Koethe.isNil_matrix_prod_of_two_by_two_matrix If M_2(J) is nil for every nil ideal J of every ring in universe u, then nilness of M_m(I) implies nilness of M_{2m}(I), using M_2(M_m(R)) ≃ M_{2m}(R)
Koethe.KotherConjecture.variants.general_matrix_of_two_by_two_matrix Documented declaration: general_matrix_of_two_by_two_matrix
Koethe.KotherConjecture.variants.general_matrix_of_matrixOver_KotherRadical The formulation of the Köthe conjecture in terms of the Köthe radical of matrix rings implies the general matrix formulation
Koethe.KotherConjecture.variants.matrixOver_KotherRadical_of_general_matrix The general matrix formulation of the Köthe conjecture implies the formulation in terms of the Köthe radical of matrix rings
Koethe.amitsur_conjecture The Amitsur Conjecture: If J is a nil ideal in R, then J[x] is a nil ideal of the polynomial ring R[x]
KomlosConjecture.komlos_conjecture.variants.zero_vectors Sanity check: with no vectors at all ($n = 0$), the empty signed sum is $0$ in every coordinate, so any constant bound holds
KummerVandiver.kummer_vandiver Kummer–Vandiver conjecture states that for every prime $p$, the class number of the maximal real subfield of $\mathbb{Q}(\zeta_p)$ is not divisible by $p$
LanderParkinSelfridge.lander_parkin_selfridge Documented declaration: lander_parkin_selfridge
LanderParkinSelfridge.lander_parkin_selfridge.variants.five_three Documented declaration: five_three
LegendreConjecture.legendre_conjecture Does there always exist at least one prime between consecutive perfect squares?
LegendreConjecture.bounded_gap_legendre Documented declaration: bounded_gap_legendre
LegendreConjecture.legendre_conjecture.ferreira_large_n Ferreira proved that the conjecture is true for sufficiently large n
LehmerMahlerMeasureProblem.mahlerMeasure The Mahler measure of f(X) is defined as ‖a‖ ∏ᵢ max(1,‖αᵢ‖), where f(X)=a(X-α₁)(X-α₂)...(X-αₙ)
LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem Let M(f) denote the Mahler measure of f
LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem.variants.best μ=M(X^10 + X^9 - X^7 - X^6 - X^5 - X^4 - X^3 + X + 1) is the best value for lehmer_mahler_measure_problem
LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem.variants.not_reciprocal If $f$ is irreducible and not reciprocal and $M(f) > 1$ then $M(f) \ge M(X^3 - X - 1)$
LehmerMahlerMeasureProblem.Polynomial.HasOddCoeffs Polynomial.HasOddCoeffs f means that all coefficients of f : Polynomial ℤ are odd
LehmerMahlerMeasureProblem.lehmer_mahler_measure_problem.variants.odd If $f$ is not reciprocal, all the coefficients of $f$ are odd and $M(f) > 1$, then $M(f) \ge M(X^2 - X - 1)$
LehmerTotient.lehmer_totient Does there exist a composite number $n > 1$ such that Euler’s totient function $\varphi(n)$ divides $n - 1$?
LeinsterGroup.IsLeinster A finite group G is a Leinster group if the sum of the orders of all its normal subgroups equals twice the group's order
LeinsterGroup.infinitely_many_leinster_groups Conjecture: Are there infinitely many Leinster groups?
LeinsterGroup.cyclic_of_perfect_is_leinster Cyclic groups of perfect number order are Leinster groups
LeinsterGroup.abelian_is_leinster_iff_cyclic_perfect An abelian group is a Leinster group if and only if it is cyclic with order equal to a perfect number
LeinsterGroup.exists_nonabelian_leinster_group Non-abelian Leinster groups exist
LeinsterGroup.dihedral_is_leinster_iff_odd_perfect The dihedral group DihedralGroup n (of order 2n) is a Leinster group if and only if n is an odd perfect number
Lemoine.lemoine_conjecture For all odd integers $n ≥ 7$ there are prime numbers $p,q$ such that $n = p+2q$
Lemoine.lemoine_conjecture_extension For all odd integers $n ≥ 9$ there are odd prime numbers $p,q,r,s$ and natural numbers $a,b$ such that $p+2q = n$, $2+pq = 2^a+r$, $2p+q = 2^b+s$
LittlewoodConjecture.littlewood_conjecture For any two real numbers
LittlewoodConjecture.padic_littlewood_conjecture For real number
LocalUniformization.hasLocalUniformization_top The trivial valuation subring of a function field is uniformized by any affine model, since its centre is the generic point
LocalUniformization.local_uniformization Local uniformization in positive characteristic
LocalUniformization.local_uniformization_of_charZero Zariski's local uniformization theorem
LocalUniformization.local_uniformization_of_trdeg_le_three Local uniformization holds in transcendence degree at most three over any field
LocalUniformization.local_uniformization_after_finite_extension Local uniformization after a finite extension
LonelyRunnerConjecture.lonely_runner_conjecture Consider $n$ runners on a circular track of unit length
LonelyRunnerConjecture.deltaTuple For an $n$-tuple of distinct integer velocities $v_1,\dots,v_n$, deltaTuple v is the maximal value of $\min_i \|t v_i\|_{\mathbb{R}/\mathbb{Z}}$ over time
LonelyRunnerConjecture.deltaGap The $n$th gap of loneliness $\delta_n$: the infimum of deltaTuple over all $n$-tuples of distinct nonzero integer velocities
LonelyRunnerConjecture.lonely_runner_conjecture.variants.tao_2017 Theorem 1.3 (Tao, 2017; arXiv:1701.02048)
LovaszPlummerConjecture.perfectMatchingCount The number of perfect matchings of G
LovaszPlummerConjecture.lovasz_plummer_conjecture The Lovász–Plummer conjecture (1970s), proved by Esperet, Kardoš, King, Král' and Norine (2011)
LovaszPlummerConjecture.lovasz_plummer_conjecture.variants.explicit The explicit bound of Esperet–Kardoš–King–Král'–Norine (2011)
LovaszPlummerConjecture.IsHamiltonianCycle A Hamiltonian cycle of G: a cycle passing through every vertex
LovaszPlummerConjecture.sheehan_conjecture Sheehan's conjecture (1977)
LovaszPlummerConjecture.sheehan_conjecture.variants.thomassen Thomassen (1998): regular graphs of large degree
LychrelNumbers.base The base (10) used for digit reversal
LychrelNumbers.rev10 The digit-reversal map $\operatorname{rev}_{10}(n)$
LychrelNumbers.IsPalindrome10 A number is a (base-10) palindrome if it equals its digit reversal
LychrelNumbers.lychrelStep One step of the Lychrel iteration: n ↦ n + rev10 n
LychrelNumbers.IsLychrel10 The number $n$ is a (base-10) Lychrel number if no iterate of the Lychrel process is a palindrome
LychrelNumbers.no_lychrel_numbers_base10 Lychrel conjecture (base 10): conjecturally, there are no Lychrel numbers in base 10
LychrelNumbers.isLychrel10_196 The first widely studied open case: whether 196 is a base-10 Lychrel number
LychrelNumbers.eventually_palindrome_base10 An equivalent formulation of no_lychrel_numbers_base10
LychrelNumbers.rev10_120 Sanity check: digit reversal of 120 is 21
LychrelNumbers.palindrome_121 Sanity check: 121 is a base-10 palindrome
LychrelNumbers.lychrelIter_56_one Sanity check: 56 → 121 in one Lychrel step
LychrelNumbers.eventually_palindrome_56 Sanity check: the Lychrel iteration at 56 reaches a palindrome
MagicSquares.exists_magic_square_squares Does there exist a $3 \times 3$ matrix such that every entry is a distinct square, and all rows, columns, and diagonals add up to the same value?
MagicSquares.exists_semi_magic_square_cubes Does there exist a $3 \times 3$ semi-magic square whose entries are all distinct positive integer cubes?
Mahler32.Ω Documented declaration: Ω
Mahler32.mahler_conjecture The Mahler Conjecture states that there are no Z-numbers
Mahler32.mahler_conjecture.variants.consequence Mahler's conjecture would follow if Ω(3/2) exceeded 1/2: a Z-number x has all fractional parts {x (3/2)^n} below 1/2, so Ω(3/2) ≤ 1/2
Mahler32.mahler_conjecture.variants.flatto_lagarias_pollington Documented declaration: flatto_lagarias_pollington
Mandelbrot.multibrotSet The Multibrot set of power n is the set of all parameters c : ℂ for which 0 does not escape to infinity under repeated application of z ↦ z ^ n + c
Mandelbrot.mandelbrotSet The Mandelbrot set is the special case of the multibrot set for n = 2
Mandelbrot.multibrotSet_eq Documented declaration: multibrotSet_eq
Mandelbrot.mandelbrotSet_eq The mandelbrot set is equivalently the set of all parameters c for which the orbit of 0 under z ↦ z ^ 2 + c does not leave the closed disk of radius two around the origin
Mandelbrot.MLC The MLC conjecture, stating that the mandelbrot set is locally connected
Mandelbrot.MLC_general_exponent A stronger version of the MLC conjecture, stating that all multibrots are locally connected
Mandelbrot.IsAttractingCycle Documented declaration: IsAttractingCycle
Mandelbrot.isAttractingCycle_z_squared_minus_one For example, 0 is part of an attracting 2-cycle of z ↦ z ^ 2 - 1
Mandelbrot.not_isAttractingCycle_z_squared_minus_two On the other hand, while 2 is part of a 1-cycle of z ↦ z ^ 2 - 2, that cycle is not attracting
Mandelbrot.no_attractingCycle_period_zero No function has an attracting cycle of period 0
Mandelbrot.density_of_hyperbolicity The density of hyperbolicity conjecture, stating that the set of all parameters c for which fun z ↦ z ^ 2 + c has an attracting cycle is dense in the Mandelbrot set
Mandelbrot.density_of_hyperbolicity_general_exponent Documented declaration: density_of_hyperbolicity_general_exponent
Mandelbrot.multibrotSet_frontier_measurable The boundary of any Multibrot set is measurable because it is closed, so it makes sense to ask about its area
Mandelbrot.volume_frontier_mandelbrotSet_eq_zero The boundary of the Mandelbrot set is conjectured to have zero area
Mandelbrot.volume_frontier_multibrotSet_eq_zero The boundary of any Multibrot set is conjectured to have zero area
MeanValueProblem.mean_value_problem Given a complex polynomial $p$ of degree $d ≥ 2$ and a complex number $z$ there is a critical point $c$ of $p$, such that $|p(z)-p(c)|/|z-c| ≤ |p'(z)|$
MeanValueProblem.mean_value_problem_leq_4 The following weaker version of the mean value problem has been proven
MeanValueProblem.mean_value_problem_of_real_roots The following tighter bound depending on the degree $d$ of the polynomial $p$ has been shown by Tischler
MeanValueProblem.mean_value_problem_of_roots_same_norm The following tighter bound depending on the degree $d$ of the polynomial $p$ has been shown by Tischler
Mersenne.Nat.GivesWagstaffPrime A Wagstaff prime is a prime number of the form $(2^p+1)/3$
Mersenne.Nat.IsSpecialForm Holds when there is exists a number k such that $p = 2^k \\pm 1$ or $p = 4^k \\pm 3$
Mersenne.catalanMersenne The Catalan-Mersenne numbers, defined recursively by $c_0 = 2$ and $c_{n+1} = 2^{c_n} - 1$
Mersenne.NewMersenneConjectureStatement A natural number p satisfies the statement of the New Mersenne Conjecture if whenever two of the following conditions hold, then all three must hold: 1
Mersenne.new_mersenne_conjecture For any odd natural number p if two of the following conditions hold, then all three must hold: 1
Mersenne.new_mersenne_conjecture_of_prime It suffices to check this conjecture for odd primes
Mersenne.new_mersenne_conjecture.variants.prime The New Mersenne Conjecture statement holds for odd primes
Mersenne.infinitely_many_mersenne_primes Are there infinitely many Mersenne primes?
Mersenne.catalans_mersenne_conjecture The first five Catalan-Mersenne numbers $c_0, \ldots, c_4$ are known to be prime
Mills.IsMills Given any real $A$, IsMills A encodes the statement that $\lfloor A^{3^n}\rfloor,n > 0$ is prime
Mills.exists' Mills' theorem (Mills, 1947)
Mills.IsMinMills For a real $A$, IsMinMills A is the smallest value satisfying IsMills A
Mills.exists_least Mills' constant
Mills.irrational Mills' constant is irrational (Saito, 2024)
Mills.lower_bound_of_RH Mills' constant lower bound (Caldwell–Cheng, 2005): assuming the Riemann hypothesis, Mills' constant begins at $1.3063778838\ldots$
ModularityConjecture.modularFormAn The n-th Fourier coefficient of a modular form (around the cusp at infinity)
ModularityConjecture.ratRed We need to reduce a rational modulo p, in practice we won't be dividing by zero since the conductor of the elliptic curve saves us
ModularityConjecture.setOfPointsModN The set of points on an elliptic curve over ZMod n
ModularityConjecture.apFintype The set of point mod n is finite
ModularityConjecture.WeierstrassCurve.ap Note that normally this is written as p + 1 - #E(𝔽ₚ), but since we don't have a point at infinity on this affine curve we only have p
ModularityConjecture.IsNormalisedEigenform Since we don't have Hecke operators yet, we define this via the q-expansion coefficients
ModularityConjecture.modularityConjecture See theorem 8.8.1 of [diamondshurman2005]
ModularityConjecture.modularity_conjecture The Modularity Theorem (formerly Shimura-Taniyama-Weil conjecture): every elliptic curve over $\mathbb{Q}$ is modular
MoserWorm.Worms The set of worms is the set of curves of length (at most) 1
MoserWorm.WormCovers The set of covers is the set of (measurable) sets that cover every worm by translation and rotation
MoserWorm.disc_mem_worm_covers A disc of radius 1 / 2 is a worm cover
MoserWorm.mosers_worm_problem Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a shape that can cover every unit-length curve?
MoserWorm.mosers_worm_problem_upper_bound There is a set of area 0.260437 that covers all worms
MoserWorm.convex_mosers_worm_problem Convex Moser's Worm Problem What is the minimal area (or greatest lower bound on the area) of a convex shape that can cover every unit-length curve?
MoserWorm.convex_mosers_worm_problem_bound_attained The minimal area of a convex shape that can cover every unit-length curve is attained
MoserWorm.convex_mosers_worm_problem_upper_bound There is a convex set of area 0.270911861 that covers all worms
MoserWorm.convex_mosers_worm_problem_lower_bound 0.232239 is a lower bound on the area of a convex set that covers all worms
MovingSofa.horizontalHallway The horizontal side of the hallway is $(-\infty, 1] \times [0, 1]$
MovingSofa.verticalHallway The vertical side of the hallway is $[0, 1] \times (-\infty, 1]$
MovingSofa.hallway The hallway is the union of its horizontal and vertical sides
MovingSofa.IsMovingSofa A connected closed set
MovingSofa.unitSquare The unit square
MovingSofa.mem_Icc_of_mem_unitSquare Coordinates of points in the unit square lie in [0,1]
MovingSofa.isMovingSofa_unitSquare The unit square $[0,1]^2$ is a valid moving sofa (with the identity motion)
MovingSofa.rotateTranslate The rigid motion that translates by $p$ and then rotates counterclockwise by $\alpha$
MovingSofa.rotateTranslate_apply rotateTranslate α p sends $q$ to $R_\alpha(q + p)$: the translation is applied first
MovingSofa.sofaOfRotateTranslatePath The sofa according to a rotation path
MovingSofa.GerversSofa.ABφθSpec Eq. 1-4 of [Ro18], which specifies the constants $A$, $B$, $\varphi$, and $\theta$ of [Ge92]
MovingSofa.gerversSofa Gerver's sofa is the sofa according to the rotation path GerversSofa.p
MovingSofa.sofaConstant The sofa constant is the maximal area of a moving sofa
MovingSofa.one_le_sofaConstant The sofa constant is at least 1, as witnessed by the unit square
MovingSofa.volume_eq_sofaConstant_iff_congruent_gerversSofa Gerver's sofa is the unique sofa that attains the sofa constant, up to a rigid motion
NoetherProblem.IsRationalExtension A rational field extension is a field extension L/K isomorphic to a field of rational functions (in some arbitrary number of indeterminates.)
NoetherProblem.rationalExtension_empty_index If the index set ι is empty, then IsRationalExtension K L ι means that K, L are isomorphic as K algebras
NoetherProblem.HasNoetherProperty Documented declaration: HasNoetherProperty
NoetherProblem.noether_problem The Noether Problem: let L be the field of rational functions in n indeterminates over K, and let G be a finite group permuting these indeterminates
NoetherProblem.noether_problem.variants.two The Noether problem has a positive solution for groups permuting two indeterminates
NoetherProblem.noether_problem.variants.three The Noether problem has a positive solution for groups permuting three indeterminates
NoetherProblem.noether_problem.variants.four The Noether problem has a positive solution for groups permuting four indeterminates
NoetherProblem.noether_problem.variants.forty_seven One can find a counterexample to the Noether Problem's claim by considering a group permuting the 47 indeterminates of a rational function field
NormalNumber.pi_normal_base_ten $\pi$ is normal in base 10
Oppermann.oppermann_conjecture.parts.i For every integer $x \ge 2$ there exists a prime between $x(x-1)$ and $x^2$
Oppermann.oppermann_conjecture.parts.ii For every integer $x \ge 2$ there exists a prime between $x^2$ and $x(x+1)$
Oppermann.oppermann_conjecture Oppermann's Conjecture: For every integer $x \ge 2$, the following hold: - There exists a prime between $x(x-1)$ and $x^2$
Oppermann.oppermann_implies_brocard Oppermann's conjecture implies Brocard's conjecture
Oppermann.oppermann_implies_legendre Oppermann's conjecture implies Legendre's conjecture
Oppermann.oppermann_conjecture.ferreira_large_x Ferreira proved that Oppermann's conjecture is true for sufficiently large x
PebblingNumberConjecture.PebbleDistribution A Pebble distribution is an assignment of zero or more pebbles to each of the vertices
PebblingNumberConjecture.NumberOfPebbles The number of pebbles of a distribution is the total number summed over all vertices
PebblingNumberConjecture.IsPebblingMove Documented declaration: IsPebblingMove
PebblingNumberConjecture.PebblePath A pebble path is a series of pebbling moves
PebblingNumberConjecture.ExistsPebblePath Indicates whether there exists a sequence of pebbling moves transforming one pebble distribution to another
PebblingNumberConjecture.IsReachable A pebble distribution B is reachable from another pebble distribution A, if there exists a sequence of pebbling moves transforming the first into the second
PebblingNumberConjecture.PebblingNumber Documented declaration: PebblingNumber
PebblingNumberConjecture.PebblingNumber_completeGraph The pebbling number of the complete graph on n vertices is n
PebblingNumberConjecture.pebbling_number_conjecture The pebbling number conjecture: the pebbling number of a Cartesian product of connected graphs is at most equal to the product of the pebbling numbers of the factors
PellNumbers.pellNumber The Pell numbers $P_n$ are defined by $P_0 = 0$, $P_1 = 1$, $P_{n+2} = 2*P_{n+1} + P_n$
PellNumbers.pellNumber_zero The initial Pell number is $P_0 = 0$
PellNumbers.pellNumber_one The second initial value is $P_1 = 1$
PellNumbers.pellNumber_two The recurrence gives $P_2 = 2$
PellNumbers.pellNumber_five The Pell number at index $5$ is $P_5 = 29$
PellNumbers.pellNumber_sq_add_pellNumber_succ_sq Similar to Fibonacci numbers, there exist numerous identities around Pell numbers, i.e. P_{2n+1} = P_n ^ 2 + P_{n+1} ^ 2
PellNumbers.coe_pellNumber_eq An explicit formula for Pell numbers, similar to Binet's formula
PellNumbers.infinite_pellNumber_primes There are infinitely many prime Pell numbers
PerfectNumbers.infinitely_many_perfect Infinitely many perfect numbers conjecture
PerfectNumbers.infinitely_many_even_perfect Infinitely many even perfect numbers conjecture
PerfectNumbers.odd_perfect_number_conjecture Odd Perfect Number Conjecture
PerfectNumbers.odd_perfect_number.lower_bound A known result: If an odd perfect number exists, it must be greater than $10^{1500}$ and must have at least 101 prime factors (including multiplicities)
PerfectNumbers.odd_perfect_number.euler_form A known result: If an odd perfect number exists, it must be of the form $p^α * m^2$ where $p$ is prime, $p \equiv 1 \pmod{4}$, $\alpha \equiv 1 \pmod{4}$, and $p \nmid m$
PierceBirkhoff.IsSemiAlgebraic Documented declaration: IsSemiAlgebraic
PierceBirkhoff.IsSemiAlgebraic₁ A set is semi-algebraic in ℝ if it can be described by a finite boolean combination of polynomial equations and inequalities
PierceBirkhoff.IsPiecewiseMvPolynomial Documented declaration: IsPiecewiseMvPolynomial
PierceBirkhoff.IsPiecewisePolynomial Documented declaration: IsPiecewisePolynomial
PierceBirkhoff.pierce_birkhoff_conjecture Documented declaration: pierce_birkhoff_conjecture
PierceBirkhoff.pierce_birkhoff_conjecture_dim_one The Pierce-Birkhoff conjecture holds for n = 1
PierceBirkhoff.pierce_birkhoff_conjecture_dim_two The Pierce-Birkhoff conjecture holds for n = 2
PierpontPrime.IsPierpontPrime A Pierpont prime is a prime of the form $2^a3^b + 1$
PierpontPrime.infinitely_many_pierpont_primes There are infinitely many Pierpont primes
PollocksConjecture.tetrahedral The $n$-th tetrahedral number: $T_n = \frac{n(n+1)(n+2)}{6}$
PollocksConjecture.NotSumOfFourTetrahedral The set of natural numbers that are not a sum of $4$ tetrahedral numbers
PollocksConjecture.pollock_tetrahedral Pollock's (tetrahedral numbers) conjecture: every integer is the sum of at most $5$ tetrahedral numbers
PollocksConjecture.pollock_tetrahedral.salzer_levine Salzer–Levine strengthening (as stated on Wikipedia/OEIS): there are exactly $241$ integers that are not a sum of $4$ tetrahedral numbers, and the largest is $343867$
PollocksConjecture.pollock_tetrahedral.ncard_exceptions As stated on Wikipedia/OEIS (A797), the set of exceptions has cardinality $241$
PolyTime.isPolyTime_primeFactorsList The integer factorization problem: Can the prime factorization of a positive integer be computed in polynomial time?
PompeiuProblem.RigidMotion A rigid motion of $\mathbb{R}^{N+1}$: an affine isometry of Euclidean space onto itself (a composition of translations and rotations/reflections)
PompeiuProblem.HasPompeiuProperty A set Ω has the Pompeiu property if the only continuous f : ℝ^{N+1} → ℝ whose integral over every rigid-motion image σ(Ω) vanishes is the zero function
PompeiuProblem.IsBall Ω is a Euclidean ball: an open metric ball of positive radius
PompeiuProblem.HasLipschitzBoundary Documented declaration: HasLipschitzBoundary
PompeiuProblem.IsAdmissibleDomain The domain hypotheses of Pompeiu's problem
PompeiuProblem.ball_not_hasPompeiuProperty The classical (easy) direction, proved by Brown–Schreiber–Taylor [BST73]: a Euclidean ball fails to have the Pompeiu property
PompeiuProblem.integral_zero_eq_zero Documented declaration: integral_zero_eq_zero
PowerfulNumbersDensity.Q $Q(x)$ is the number of powerful integers $n$ with $1 \le n \le x$
PowerfulNumbersDensity.A The leading constant $\zeta(3/2)/\zeta(3) = 2.173\ldots$
PowerfulNumbersDensity.B The second-order constant $\zeta(2/3)/\zeta(2)$, where $\zeta(2/3)$ is given by analytic continuation
PowerfulNumbersDensity.asymptotic_erdos_szekeres Documented declaration: asymptotic_erdos_szekeres
PowerfulNumbersDensity.asymptotic_bateman_grosswald Bateman and Grosswald [BG58] proved the sharper asymptotic $$Q(x) = \frac{\zeta(3/2)}{\zeta(3)} x^{1/2} + \frac{\zeta(2/3)}{\zeta(2)} x^{1/3} + O(x^{1/6}).$$
PowerfulNumbersDensity.error_term_improvement Can the exponent $1/6$ in the error term of the Bateman–Grosswald asymptotic be improved unconditionally?
PrimeTriplets.prime_triplets Are there infinitely many tuples of three consecutive primes $(p, q, r)$ such that $r - p = 6$?
PrimesAndPerfectSquares.infinite_prime_sq_add_one Are there infinitely many primes $p$ such that $p - 1$ is a perfect square?
QuasiperfectNumbers.Quasiperfect A number is quasiperfect if the sum of its divisors is equal to $2n + 1$
QuasiperfectNumbers.exists_quasiperfect Quasiperfect Numbers Conjecture
RamanujanTau.ramanujan_petersson The Ramanujan-Petersson conjecture: $|\tau(p)| \le 2 p^{11/2}$ for primes $p$
RamanujanTau.lehmer_ramanujan_tau Lehmer's conjecture: $\tau(n) \ne 0$ for all $n > 0$
RamseyNumbers.IsGraphRamsey Documented declaration: IsGraphRamsey
RamseyNumbers.IsGraphRamsey.succ Monotonicity in the number of vertices
RamseyNumbers.IsGraphRamsey.symm Symmetry in the clique / independent set sizes
RamseyNumbers.classicalRamsey_eq_sInf The shared Ramsey number agrees with the clique-free formulation
RamseyNumbers.ramsey_number_five_five The open problem: determine the Ramsey number $R(5,5)$
RamseyNumbers.ramsey_number_five_five_lower_bound Lower bound $43 \le R(5,5)$, equivalently: there exists a graph on $42$ vertices with no $5$-clique and no independent set of size $5$
RamseyNumbers.ramsey_number_five_five_upper_bound Upper bound $R(5,5) \le 46$, i.e. every graph on $46$ vertices contains a $5$-clique or an independent set of size $5$
RamseyNumbers.ramsey_number_three_three $R(3,3) = 6$
RamseyNumbers.ramsey_number_three_four $R(3,4) = 9$
RamseyNumbers.ramsey_number_three_five $R(3,5) = 14$
RamseyNumbers.ramsey_number_three_six $R(3,6) = 18$
RamseyNumbers.ramsey_number_three_seven $R(3,7) = 23$
RamseyNumbers.ramsey_number_three_eight $R(3,8) = 28$
RamseyNumbers.ramsey_number_three_nine $R(3,9) = 36$ (Grinstead–Roberts)
RamseyNumbers.ramsey_number_four_four The diagonal Ramsey number $R(4,4) = 18$
RamseyNumbers.ramsey_number_four_five $R(4,5) = 25$ (McKay–Radziszowski)
RamseyNumbers.ramsey_number_six_six The diagonal Ramsey number $R(6,6)$ is unknown
RamseyNumbers.ramsey_number_six_six_lower_bound Lower bound $102 \le R(6,6)$, equivalently: there exists a graph on $101$ vertices with no $6$-clique and no independent set of size $6$
RamseyNumbers.ramsey_number_six_six_upper_bound Upper bound $R(6,6) \le 165$, i.e. every graph on $165$ vertices contains a $6$-clique or an independent set of size $6$
RationalDistanceProblem.rational_distance_problem Does there exist a point in the plane at rational distance from all four vertices of the unit square?
RegularPrimes.IsRegularPrime A natural prime number p is regular if p is coprime with the order of the class group of the p-th cyclotomic field
RegularPrimes.not_isRegularPrime_37 The prime 37 is not a regular prime
RegularPrimes.regularPrimes The set of regular primes
RegularPrimes.irregularPrimes The set of irregular primes
RegularPrimes.small_regular_primes The primes 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, and 31 are regular
RegularPrimes.isRegularPrime_iff_Bernoulli Kummer's criterion
RegularPrimes.infinitude_of_irregularprimes The set of irregular primes is infinite
RegularPrimes.RegularPrimeConjecture Conjecture: The set of regular primes is infinite
RegularPrimes.regularprime_conjecture Conjecture: The set of regular primes is infinite
RepunitPrime.repunit10 The decimal repunit of length $n$, defined as $1 + 10 + \cdots + 10^{n-1}$
RepunitPrime.infinitely_many_repunit_primes Wikipedia's repunit article states: "It has been conjectured that there are infinitely many repunit primes."
AlgebraicGeometry.Scheme.HasResolution Documented declaration: HasResolution
AlgebraicGeometry.Scheme.hasResolution_of_smooth A scheme that is already smooth over the base is its own resolution
AlgebraicGeometry.Scheme.surjective_of_birationalOver_id A proper birational morphism onto a scheme is surjective
AlgebraicGeometry.resolution_of_singularities Resolution of singularities in positive characteristic
AlgebraicGeometry.resolution_of_singularities_of_charZero Hironaka's theorem
AlgebraicGeometry.resolution_of_singularities_of_topologicalKrullDim_le_three Resolution of singularities holds in dimension at most three over any perfect field
AlgebraicGeometry.exists_not_hasResolution_of_not_perfectField Resolution of singularities fails over imperfect fields
RiemannZetaValues.irrational_seven $\zeta(7)$ is irrational
RiemannZetaValues.irrational_nine $\zeta(9)$ is irrational
RiemannZetaValues.irrational_eleven $\zeta(11)$ is irrational
RiemannZetaValues.irrational_odd $\zeta(2n + 1)$ is irrational for any $n\in\mathbb{N}^{+}$
RiemannZetaValues.irrational_three $\zeta(3)$ is irrational
RiemannZetaValues.infinite_irrational_at_odd There are infinitely many $\zeta(2n + 1)$, $n \in \mathbb{N}$, that are irrational
RiemannZetaValues.exists_irrational_of_five_seven_nine_eleven At least one of $\zeta(5), \zeta(7), \zeta(9)$ or $\zeta(11)$ is irrational
RudinsConjecture.Q $Q(N; q, a)$ is the number of perfect squares among the first $N$ terms of the arithmetic progression $\{q n + a : 0 \le n < N\}$
RudinsConjecture.IsNontrivial A pair $(q, a)$ describes a non-trivial arithmetic progression if $q, a \ge 1$, $\gcd(q, a) = 1$, and $(q, a) \neq (1, 1)$
RudinsConjecture.Qmax $Q(N) = \max Q(N; q, a)$, the largest number of perfect squares occurring among the first $N$ terms of any non-trivial arithmetic progression
RudinsConjecture.isNontrivial_24_1 Sanity check: the progression $24 n + 1$ is non-trivial
RudinsConjecture.Q_zero Sanity check: there are no squares among the first 0 terms of any progression
RudinsConjecture.Q_six_twentyfour_one Documented declaration: Q_six_twentyfour_one
RudinsConjecture.rudins_conjecture Rudin's conjecture
RudinsConjecture.rudins_conjecture_strong A stronger form of Rudin's conjecture: for every $N \ge 6$, the arithmetic progression $24 n + 1$ attains the maximum $Q(N)$
RudinsConjecture.rudins_conjecture_unique The strongest form of Rudin's conjecture also asserts uniqueness: for $N \ge 6$, any non-trivial arithmetic progression attaining the maximum $Q(N)$ has common difference $24$
SatoTateConjecture.cmJInvariants The thirteen rational CM $j$-invariants, corresponding respectively to the imaginary quadratic orders of discriminants $-3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163$
SatoTateConjecture.HasCM An elliptic curve $E$ over $\mathbb{Q}$ has complex multiplication over $\overline{\mathbb{Q}}$ if its $j$-invariant belongs to the set of thirteen rational CM $j$-invariants
SatoTateConjecture.normalisedAp The normalized coefficient $a_p(E)/(2\sqrt{p})$, using the point-counting definition from ModularityConjecture
SatoTateConjecture.satoTateCDF The cumulative distribution function of the Sato–Tate measure
SatoTateConjecture.satoTateMeasure For $a \le b$, the mass assigned to $[a,b]$ by the Sato–Tate distribution
SatoTateConjecture.satoTateIntegral For $a \le b$, the mass assigned to $[a,b]$ by the Sato–Tate distribution, written in its usual integral form $$ \frac{2}{\pi}\int_a^b \sqrt{1-x^2}\,dx. $$
SatoTateConjecture.satoTateMeasure_eq_satoTateIntegral Sanity check: for $-1 \le a \le b \le 1$, the closed-form interval mass satoTateMeasure agrees with the integral form satoTateIntegral
SatoTateConjecture.primeCountInInterval The number of primes $p < N$ for which the normalized coefficient belongs to $[a,b]$
SatoTateConjecture.satoTate_conjecture The Sato–Tate conjecture: for a non-CM elliptic curve
Schanuel.schanuel_conjecture Documented declaration: schanuel_conjecture
Schinzel.schinzel_conjecture Schinzel conjecture (H hypothesis) If a finite set of polynomials
ScholzConjecture.scholz_conjecture The Scholz conjecture, also known as the Scholz-Brauer conjecture, asserts that for every positive integer $n$, the addition-chain length of $2^n - 1$ is at most $n - 1 + \ell(n)$
ScholzConjecture.three_notMem_additionChainSteps_seven 7 is the first value where the doubling bound is not sharp: it gives ℓ(7) ≥ 3, and no four-entry chain ends at 7
ScholzConjecture.additionChainLength_first_values The first few values of $\ell(n)$
Selfridge.IsSelfridge A number p satisfies the Selfridge condition if 1
Selfridge.IsPseudoSelfridge A number p satisfies the Pseudo Selfridge condition if 1
Selfridge.selfridge_conjecture PSW conjecture (Selfridge's test) Let $p$ be an odd number, with $p \equiv \pm 2 \pmod{5}$, $2^{p-1} \equiv 1 \pmod{p}$ and $F_{p+1} \equiv 0 \pmod{p}$, then $p$ is a prime number
Selfridge.selfridge_conjecture.variants.exist_pseudo_counterexample Selfridge's test variant: Let $p$ be an odd number, with $p \equiv \pm 1 \pmod{5}$, $2^{p-1} \equiv 1 \pmod{p}$ and $F_{p-1} \equiv 0 \pmod{p}$, then $p$ is a prime number
Selfridge.selfridge_conjecture.variants.pseudo_counterexample Selfridge's test variant: Let $p$ be an odd number, with $p \equiv \pm 1 \pmod{5}$, $2^{p-1} \equiv 1 \pmod{p}$ and $F_{p-1} \equiv 0 \pmod{p}$, then $p$ is a prime number
Selfridge.selfridge_conjecture.variants.pseudo_counterexample' Selfridge's test variant: Let $p$ be an odd number, with $p \equiv \pm 1 \pmod{5}$, $2^{p-1} \equiv 1 \pmod{p}$ and $F_{p-1} \equiv 0 \pmod{p}$, then $p$ is a prime number
Selfridge.fermatFactors OEIS A46052 The number of distinct prime factors of nth Fermat number
Selfridge.selfridge_seq_conjecture Selfridge conjectured that the number of prime factors of the n-th Fermat number does not grow monotonically in $n$
Selfridge.selfridge_seq_conjecture.variants.sufficient_condition Selfridge conjectured that the number of prime factors of the n-th Fermat number does not grow monotonically in $n$
Sendov.Polynomial.IsSendov The predicate that a polynomial satisfies the hypotheses of Sendov's conjecture
Sendov.Nat.SatisfiesSendovConjecture SatisfiesSendovConjecture n states that Sendov's conjecture is true for every polynomial of degree n
Sendov.sendov_conjecture.variants.le_eight Sendov's conjecture states that for a polynomial
Sendov.sendov_conjecture.variants.eventually_true Sendov's conjecture states that for a polynomial
SerreMultiplicityConjectures.supportDim_add_supportDim_le_ringKrullDim Dimension inequality
SerreMultiplicityConjectures.intersectionMultiplicity_nonneg Non-negativity
SerreMultiplicityConjectures.intersectionMultiplicity_eq_zero_of_lt Vanishing
SerreMultiplicityConjectures.intersectionMultiplicity_pos Positivity conjecture
SerreMultiplicityConjectures.intersectionMultiplicity_pos_quotient Positivity conjecture, in the form stated on Wikipedia
SerreMultiplicityConjectures.intersectionMultiplicity_pos_of_equalCharacteristic Serre proved the positivity conjecture when $R$ is of equal characteristic, that is, when $R$ contains a field [Se00]
SerreMultiplicityConjectures.intersectionMultiplicity_pos_of_unramified Documented declaration: intersectionMultiplicity_pos_of_unramified
SidorenkoConjecture.sidorenko_conjecture Sidorenko's conjecture (1993)
SidorenkoConjecture.sidorenko_conjecture_graphon Sidorenko's conjecture for graphons (1993)
SidorenkoConjecture.sidorenko_tree_graphon Case: H is a tree (Sidorenko 1993, graphon version)
SidorenkoConjecture.sidorenko_even_cycle_graphon Case: H = C_{2k} is an even cycle (Sidorenko 1993, graphon version)
SidorenkoConjecture.sidorenko_completeBipartiteGraph_graphon Case: H = K_{a,b} is a complete bipartite graph (Sidorenko 1993, graphon version)
SidorenkoConjecture.tournament_anti_sidorenko_trees_conjecture Tournament Anti-Sidorenko (TAS) Trees Conjecture
SidorenkoConjecture.tournament_anti_sidorenko_trees_conjecture_tournamenton Tournament Anti-Sidorenko (TAS) Trees Conjecture (Tournamenton limit version)
SidorenkoConjecture.tournament_anti_sidorenko_single_even_degree_tree TAS Trees Conjecture: Trees with a single even-degree vertex
SidorenkoConjecture.IsSpider234 The $(2,3,4)$-spider tree
SidorenkoConjecture.tournament_anti_sidorenko_spider234 TAS Trees Conjecture: The $(2,3,4)$-spider tree (Solved case)
SidorenkoConjecture.sidorenko_K2 Case H = K_2 (single edge): Sidorenko's inequality holds trivially with equality
SidorenkoConjecture.edgeCount_completeBipartiteGraph_fin_two edgeCount of K_{2,2} (complete bipartite graph on Fin 2 + Fin 2) is 4
SidorenkoConjecture.homCount_completeGraph_fin_two_eq_two_mul_card_edgeFinset Homomorphism count of K_2 into G equals 2 · #edgeFinset
SidorenkoConjecture.homCount_completeBipartiteGraph_fin_two_eq_sum_inter_sq The Hom(K_{2,2}, G) decomposition
SidorenkoConjecture.sum_inter_card_eq_sum_degree_sq K_{2,2} count via a re-indexed sum
SidorenkoConjecture.sidorenko_K22 Case H = K_{2,2} (four-cycle, also called C_4): Sidorenko's conjecture holds, by Cauchy–Schwarz
SidorenkoConjecture.sidorenko_K22_both_sides_bounded Consequence of homDensity_le_one: both sides of the Sidorenko K_{2,2} inequality are bounded above by 1
SidorenkoConjecture.edgeCount_eq_zero_of_subsingleton If the domain of H has at most one vertex, H has no edges
SidorenkoConjecture.homCount_eq_of_subsingleton If the domain of H has at most one vertex, every function V → W is a homomorphism H →g G
SidorenkoConjecture.sidorenko_tree_subsingleton Base case of Sidorenko for trees
SidorenkoConjecture.sidorenko_tree Case: H is a tree (Sidorenko 1993)
SidorenkoConjecture.sidorenko_conjecture.variants.complete_bipartite Case: complete bipartite graphs K_{s,t} (Sidorenko 1993)
SidorenkoConjecture.sidorenko_conjecture.variants.even_cycle Case: even cycles C_{2k} (Sidorenko 1993)
SidorenkoConjecture.sidorenko_conjecture.variants.path_blakley_roy Case: paths, the Blakley–Roy inequality (1965)
SidorenkoConjecture.homCount_star_eq_sum_degree_pow Homomorphism count of the star K_{1,m}
SidorenkoConjecture.card_edgeFinset_star Edge count of the star K_{1,m}
SidorenkoConjecture.sidorenko_conjecture.variants.star Case: stars K_{1,m} (Sidorenko)
SidorenkoConjecture.sidorenko_conjecture.variants.non_bipartite_necessary Bipartiteness is necessary: the triangle is not a Sidorenko graph
SierpinskiNumber.selfridge_78557 Selfridge proved in 1962 that 78557 is a Sierpiński number by showing that all numbers of the form
SierpinskiNumber.selfridge_conjecture The Sierpiński problem (Selfridge's conjecture)
SierpinskiNumber.prime_sierpinski_problem The prime Sierpiński problem
SierpinskiNumber.extended_sierpinski_problem The extended Sierpiński problem
Singmaster.solutions The set of pairs (n, k) representing the solutions to the equation Nat.choose n k = t for a given t, under the constraint 1 ≤ k < n
Singmaster.singmaster Singmaster's conjecture: the number of times any number $t > 1$ appears in Pascal's triangle is bounded
SixStandardDeviations.six_standard_deviations Documented declaration: six_standard_deviations
SixStandardDeviations.six_standard_deviations.variants.zero_sets Sanity check: with no points and no sets ($n = 0$), the empty colouring works and the bound $6\sqrt{0} = 0$ holds vacuously
SmallCohenMacaulayModules.isSmallCohenMacaulay_iff_of_krullDimLE_zero Over a local ring of Krull dimension zero, the small Cohen-Macaulay modules are exactly the nonzero finitely generated modules
SmallCohenMacaulayModules.exists_isSmallCohenMacaulay The small Cohen-Macaulay modules conjecture
SmallCohenMacaulayModules.exists_isSmallCohenMacaulay.variants.ringKrullDim_le_two The conjecture holds in dimension at most $2$
SmallCohenMacaulayModules.isBalancedBigCohenMacaulay_iff_isSmallCohenMacaulay A finitely generated module is a balanced big Cohen-Macaulay module if and only if it is a small Cohen-Macaulay module [Ho17]
SmallCohenMacaulayModules.exists_isBalancedBigCohenMacaulay Existence of balanced big Cohen-Macaulay modules
SnakeInBox.Hypercube A graph on the power set of Fin n, where two sets are adjacent if they differ by a single element
SnakeInBox.IsSnakeInGraphOfLength A subgraph G' is a 'snake' of length k in graph G if it is an induced path of length k
SnakeInBox.LongestSnakeInGraph The length of the longest induced path (or 'snake') in a graph G
SnakeInBox.LongestSnakeInTheBox The length of the longest snake for the Hypercube n graph
SnakeInBox.IsCoilInGraphOfLength A subgraph G' is a 'coil' of length k in graph G if it is an induced cycle of length k
SnakeInBox.LongestCoilInGraph The length of the longest induced cycle (or 'coil') in a graph G
SnakeInBox.LongestCoilInTheBox The length of the longest coil for the Hypercube n graph
SnakeInBox.snake_zero_zero The longest snake in the $0$-dimensional cube, i.e. the cube consisting of one point, is zero, since there only is one induced path and it is of length zero
SnakeInBox.coil_zero_zero The longest coil in the $0$-dimensional cube is zero, since it contains no cycles
SnakeInBox.snake_small_dimensions The maximum length for the snake-in-the-box problem is known for dimensions zero through eight; it is $0, 1, 2, 4, 7, 13, 26, 50, 98$
SnakeInBox.coil_small_dimensions The maximum length for the coil-in-the-box problem is known for dimensions zero through eight; it is $0, 0, 4, 6, 8, 14, 26, 48, 96$
SnakeInBox.snake_dim_nine For dimension $9$, the length of the longest snake in the box is not known
SnakeInBox.snake_dim_nine_lower_bound The best length found so far for dimension nine is 190
SnakeInBox.snake_upper_bound For $n \geq 1$, an upper bound on the length of the longest snake in a box is $2^{n-1}$ (see [DK67, Theorem B])
SnakeInBox.coil_upper_bound For $n \geq 2$, an upper bound on the maximal length of the longest coil in a box is given by $$ 1 + 2^{n-1}\frac{6n}{6n + \frac{1}{6\sqrt{6}}n^{\frac 1 2} - 7} $$ (see [Ze97])
SolitaryNumber.Friendly Two positive integers
SolitaryNumber.IsSolitary A positive integer $n$ is solitary if every friend of $n$ is equal to $n$, i.e. its abundancy class is the singleton $\{n\}$
SolitaryNumber.is_ten_solitary Is 10 a solitary number?
SolitaryNumber.infinite_club_exists Existence of an infinite club
SparseRuler.marks A ruler is described by its list of segment lengths (gaps) g, so that its marks are the partial sums
SparseRuler.IsComplete A ruler is complete (a perfect ruler) if its marks form a difference basis for $\{0, 1, \ldots, L\}$, i.e. every distance $k \le L$ is the difference of two marks
SparseRuler.IsMinimal A complete ruler is minimal if no complete ruler of the same length $L$ has fewer marks (equivalently, fewer segments)
SparseRuler.IsMaximal A complete ruler is maximal if no complete ruler with the same number of marks (equivalently, the same number of segments) has greater length
SparseRuler.IsOptimal A ruler is optimal if it is both minimal and maximal
SparseRuler.wichmannGaps The Wichmann ruler
SparseRuler.wichmannGaps_length The Wichmann ruler $W(r, s)$ has $4r + s + 2$ segments, hence $4r + s + 3$ marks [Wi63]
SparseRuler.wichmannGaps_sum The Wichmann ruler $W(r, s)$ has length $4r(r + s + 2) + 3(s + 1)$ [Wi63]
SparseRuler.wichmann_conjecture Wichmann's conjecture on optimal rulers
SquarePacking.Square A square of a particular side length as a subset of the Euclidean plane
SquarePacking.UnitSquare The unit square as a subset of the Euclidean plane
SquarePacking.Circle A circle of a particular radius as a subset of the Euclidean plane
SquarePacking.UnitCircle The unit circle as a subset of the Euclidean plane
SquarePacking.Packing A structure representing a packing of n isometric embeddings of a set s inside a (presumably larger) set S
SquarePacking.circle_zero The degenerate circle is empty
SquarePacking.eleven_square_packing_in_square_bound Eleven unit squares can be packed into a square of side length < 3.877084
SquarePacking.least_eleven_square_packing_in_square What is the smallest square that can contain 11 unit squares?
SquarePacking.seventeen_square_packing_in_square_bound Seventeen unit squares can be packed into a square of side length < 4.6756
SquarePacking.least_seventeen_square_packing_in_square What is the smallest square that can contain 17 unit squares?
SquarePacking.three_square_packing_in_circle_bound Three unit squares can be packed into a circle of radius $(5 \sqrt{17}) / 16 \approx 1.288$
SquarePacking.twenty_one_circle_packing_in_square_bound Twenty-one unit circles can be packed into a square of side length < 9.359
SquarePacking.least_twenty_one_circle_packing_in_square What is the smallest square that can contain 21 unit circles?
SquarePacking.fifteen_circle_packing_in_circle_bound Fifteen unit circles can be packed into a circle of radius $1 + \sqrt{6 + 2/\sqrt{5} + 4 \sqrt{1 + 2/\sqrt{5}}} \approx 4.521$
SquarePacking.least_fifteen_circle_packing_in_circle What is the smallest circle that can contain 15 unit circles?
SteinerSystems.SteinerSystem An $S(t, k, n)$-Steiner system is a collection of $k$-element subsets (called blocks) of $\{0, \ldots, n-1\}$ such that every $t$-element subset is contained in exactly one block
SteinerSystems.LargeSteinerSystemWitness A constructive witness for a large Steiner system: concrete values of $n$, $k$, $t$ satisfying $n > k > t > 5$, $t < 10$, $n < 200$, together with an explicit Steiner system
SteinerSystems.large_steiner_systems Construct an $S(t, k, n)$-Steiner system with $n > k > t > 5$, $t < 10$, and $n < 200$
SteinerSystems.fano_plane Sanity check: the Fano plane is an $S(2, 3, 7)$-Steiner system
SteinerSystems.steiner_system_5_6_12 Existence of $S(5, 6, 12)$: The small Witt design
SteinerSystems.steiner_system_5_8_24 Existence of $S(5, 8, 24)$: The large Witt design
SteinerSystems.infinitely_many_steiner_t4 There are infinitely many Steiner systems with $t = 4$
SteinerSystems.infinitely_many_steiner_t5 There are infinitely many Steiner systems with $t = 5$
SumOfThreeCubes.IsSumOfThreeCubes The predicate that n : R is a sum of three cubes
SumOfThreeCubes.isSumOfThreeCubesRat_any Any rational number is a sum of three rational cubes
SumOfThreeCubes.isSumOfThreeCubes_iff_mod_9 An integer n : ℤ can be written as a sum of three cubes (of integers) if and only if n is not 4 or 5 mod 9
Superperfect.PerfectFor A positive integer $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$ where $\sigma^m$ is the $m$-th iterate of $σ$
Superperfect.twoFivePerfect There does not exist a $(2,5)$-perfect number
GottschalkSurjunctivity.shift The left shift of x : G → A by g : G, defined by (shift g x)(h) = x(g⁻¹ * h)
GottschalkSurjunctivity.IsShiftEquivariant A map τ : (G → A) → (G → A) is equivariant (with respect to the left shift action) if τ(shift g x) = shift g (τ x) for all g : G and x : G → A
GottschalkSurjunctivity.IsSurjunctive A group G is surjunctive if for every finite nonempty type A, every injective, continuous, shift-equivariant map (G → A) → (G → A) is also surjective
GottschalkSurjunctivity.gottschalk_surjunctivity_conjecture Gottschalk's surjunctivity conjecture (1973): every group is surjunctive
GottschalkSurjunctivity.isSurjunctive_of_finite Every finite group is surjunctive
TarskiExponentialFunctionProblem.isDecidable_completeTheory_real_orderedField Tarski's theorem
TarskiExponentialFunctionProblem.tarski_exponential_function_problem Tarski's exponential function problem
TarskiExponentialFunctionProblem.RealSchanuelConjecture Documented declaration: RealSchanuelConjecture
TarskiExponentialFunctionProblem.realSchanuelConjecture_of_schanuelConjecture Schanuel's conjecture for complex numbers, as stated in Schanuel.schanuel_conjecture, implies its real version
TarskiExponentialFunctionProblem.isDecidable_completeTheory_realExp_of_realSchanuelConjecture Macintyre–Wilkie
Taxicab.IsTaxicabFor' Documented declaration: IsTaxicabFor'
Taxicab.taxicab_1729 $1729$ is a possible taxicab number for $k=3, m=2, n=2$
Taxicab.IsTaxicabFor $x$ is a taxicab number if it is the smallest number that can be expressed as a sum of $m$ positive $k$-th powers in at least $n$ distinct ways
Taxicab.taxicab_4 $4$ is the taxicab number for $k=1, m=2, n=2$
Taxicab.taxicab_for_5_2_2 Taxicab number for $k=5$, $m=2$, and $n=2$ is not known
Taxicab.taxicab_for_5_2_n Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$
Toronto.TorontoSpace A Toronto space is a topological space which is homeomorphic to all of its subspaces of same cardinality
Toronto.Finite.torontoSpace Every finite space is Toronto, since the only subspace with same cardinality is the space itself
Toronto.DiscreteTopology.of_t2_of_torontoSpace Any T2, Toronto space is discrete
Transcendental.exp_add_pi_transcendental $e + \pi$ is transcendental
Transcendental.exp_mul_pi_transcendental $e\pi$ is transcendental
Transcendental.exp_pow_pi_sq_transcendental $e^{\pi^2}$ is transcendental
Transcendental.exp_exp_transcendental $e^e$ is transcendental
Transcendental.pi_pow_exp_transcendental $\pi^e$ is transcendental
Transcendental.pi_pow_sqrt_two_transcendental $\pi^{\sqrt{2}}$ is transcendental
Transcendental.pi_pow_pi_transcendental $\pi^{\pi}$ is transcendental
Transcendental.pi_pow_pi_pow_pi_transcendental $\pi^{\pi^{\pi}}$ is transcendental
Transcendental.pi_pow_pi_pow_pi_pow_pi_transcendental $\pi^{\pi^{\pi^\pi}}$ is transcendental
Transcendental.pi_pow_pi_pow_pi_pow_pi_not_integer $\pi^{\pi^{\pi^\pi}}$ is not an integer
Transcendental.rlog_pi_transcendental $\log(\pi)$ is transcendental
Transcendental.rlog_rlog_two_transcendental $\log(\log(2))$ is transcendental
Transcendental.sin_exp_transcendental $\sin(e)$ is transcendental
Transcendental.transcendental_eulerMascheroniConstant_or_gompertzConstant At least one of the Euler–Mascheroni constant $\gamma$ and the Gompertz constant $\delta$ is transcendental
Transcendental.transcendental_catalanConstant The Catalan constant $G$ is transcendental
Transcendental.transcendental_gompertzConstant The Gompertz constant $\delta$ is transcendental
Transcendental.transcendental_gamma_one_div_two $\Gamma(1/2)$ is transcendental
Transcendental.transcendental_gamma_one_div_three $\Gamma(1/3)$ is transcendental
Transcendental.transcendental_gamma_one_div_four $\Gamma(1/4)$ is transcendental
Transcendental.transcendental_gamma_one_div_six $\Gamma(1/6)$ is transcendental
Transcendental.transcendental_gamma_one_div $\Gamma(1/n)$ for n ≥ 2 is transcendental
TwinPrimes.twin_primes The twin prime question
UnionClosed.union_closed For every finite union-closed family of sets, other than the family containing only the empty set, there exists an element that belongs to at least half of the sets in the family
UnionClosed.union_closed.variants.yu Yu [Yu23] showed that the union-closed sets conjecture holds with a constant of approximately 0.38234 instead of 1/2
UnionClosed.union_closed.variants.univ_card Vuckovic and Zivkovic [Vu17] showed that the union-closed sets conjecture holds for set families whose universal set has cardinality at most 12
UnionClosed.union_closed.variants.family_card Roberts and Simpson [Ro10] showed that the union-closed sets conjecture holds for set families of size at most 46
UnionClosed.union_closed.variants.univ_card_two We can show the union-closed sets conjecture is true for the case where the universal set has cardinality 2, by brute force
UnionClosed.union_closed.variants.singleton_mem We can show the union-closed sets conjecture is true for the case where the set family contains some singleton
UnionClosed.union_closed.variants.sharpness The union-closed sets conjecture is sharp in the sense that if we replace the constant 1/2 with any larger constant, then the conjecture fails
UnionClosed.union_closed.variants.cardinality_even_of_union_closed_tight Documented declaration: cardinality_even_of_union_closed_tight
VaughtConjecture.numberOfCountableModels The number of countable models of some L-Theory T up to isomorphism
VaughtConjecture.vaught_conjecture The Vaught conjecture states that for a countable language L and a complete L-Theory T the number of countable models of T (up to isomorphism) is finite
VizingConjecture.vizing_conjecture Vizing's conjecture (1968)
VizingConjecture.vizing_conjecture.variants.clark_suen The Clark–Suen inequality (2000)
VizingConjecture.vizing_conjecture.variants.suen_tarr The Suen–Tarr inequality (2012)
VizingConjecture.vizing_conjecture.variants.dominationNumber_eq_one Vizing's conjecture when γ(H) = 1
QuadraticAlgebra.discr_rat_of_modEq_one The discriminant of ℚ[√d] for d ≥ 2 squarefree congruent to 1 mod 4 is d
QuadraticAlgebra.discr_rat_of_not_modEq_one The discriminant of ℚ[√d] for d ≥ 2 squarefree not congruent to 1 mod 4 is 4 * d
Algebra.exists_quadraticAlgebra_of_isQuadraticExtension A quadratic algebra L over a field K is isomorphic to the explicit quadratic algebra QuadraticAlgebra K a b for some a b : K
Algebra.isQuadraticExtension_iff_exists_quadraticAlgebra An algebra L is quadratic over a field K iff it is isomorphic to the explicit quadratic algebra QuadraticAlgebra K a b for some a b : K
NumberField.exists_quadraticAlgebra_of_isQuadraticExtension A quadratic number field K is isomorphic to the explicit quadratic field QuadraticAlgebra ℚ d 0 for some squarefree d : ℤ not equal to 1
NumberField.isQuadraticExtension_iff_exists_quadraticAlgebra A number field K is quadratic iff it is isomorphic to the explicit quadratic field QuadraticAlgebra ℚ d 0 for some squarefree d : ℤ not equal to 1
NumberField.isFundamentalDiscr_iff_exists_discr_quadraticAlgebra An integer D is a fundamental discriminant iff it is the discriminant of the explicit quadratic field QuadraticAlgebra ℚ d 0 for some squarefree d : ℤ not equal to 1
NumberField.isFundamentalDiscr_iff_exists_discr_numberField An integer D is a fundamental discriminant iff it is the discriminant of some number field
WallSunSun.exists_isWallSunSunPrime A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \equiv 1 \pmod{p^2}$, where $L_p$ is the $p$-th Lucas number
WallSunSun.infinite_isWallSunSunPrime A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \equiv 1 \pmod{p^2}$, where $L_p$ is the $p$-th Lucas number
WallSunSun.infinite_isWallSunSunPrime_of_disc_eq_varying_parameters An earlier formulation of infinite_isWallSunSunPrime_of_disc_eq, which chose the Lucas parameters $(a, b)$ separately for every prime $p$, was degenerate: it is provable
WallSunSun.infinite_isWallSunSunPrime_of_disc_eq Let $K$ be a real quadratic field of discriminant $D$ and let $\varepsilon$ be a fundamental unit of $K$
WieferichMirimanoffPrime.exists_isWieferichPrime_and_isMirimanoffPrime Can a prime $p$ satisfy $2^{p-1} \equiv 1 \pmod{p^2}$ and $3^{p-1} \equiv 1 \pmod{p^2}$ simultaneously?
WieferichMirimanoffPrime.isMirimanoffPrime_iff Are $11$ and $1006003$ the only Mirimanoff primes?
WieferichMirimanoffPrime.isMirimanoffPrime_and_not_isWieferichPrime_11 The prime $11$ is a Mirimanoff prime but not a Wieferich prime
WieferichMirimanoffPrime.isMirimanoffPrime_and_not_isWieferichPrime_1006003 The prime $1006003$ is a Mirimanoff prime but not a Wieferich prime
WieferichMirimanoffPrime.not_isMirimanoffPrime_two Neither $2$ nor $3$ is a Mirimanoff prime, so no hypothesis excluding them is needed in the statement of the problem
WieferichMirimanoffPrime.not_isWieferichPrime_three The prime $3$ is not a Wieferich prime
WieferichPrime.infinite_isWieferichPrime There are infinitely many Wieferich primes
WieferichPrime.isWieferichPrime_iff Are $1093$ and $3511$ the only Wieferich primes?
WieferichPrime.infinite_isWieferichPrimeBase For any given integer $a > 0$, are there infinitely many primes $p$ such that $a^{p-1} \equiv 1 \pmod{p^2}$?
WieferichPrime.exists_isWieferichPrimeBase_47 Are there any Wieferich primes to base $47$?
WieferichPrime.isWieferichPrime_1093 The prime $1093$ is a Wieferich prime: $2^{1092} \equiv 1 \pmod{1093^2}$
WieferichPrime.isWieferichPrime_3511 The prime $3511$ is a Wieferich prime: $2^{3510} \equiv 1 \pmod{3511^2}$
WieferichPrime.not_isWieferichPrime_two The prime $2$ is not a Wieferich prime, so no hypothesis excluding it is needed
WilsonPrime.IsWilsonPrime A Wilson prime is a prime $p$ such that $p^2 \mid (p-1)!+1$
WilsonPrime.infinitely_many_wilson_primes There are infinitely many Wilson primes
WilsonPrime.isWilsonPrime_five The prime $5$ is a Wilson prime
WilsonPrime.isWilsonPrime_thirteen The prime $13$ is a Wilson prime
WilsonPrime.not_isWilsonPrime_one The primality condition excludes $1$, which satisfies the divisibility condition alone
WilsonPrime.not_isWilsonPrime_seven The prime $7$ is not a Wilson prime
WolstenholmePrime.IsWolstenholmePrime A prime $p > 7$ is called a Wolstenholme prime if $\binom{2p-1}{p-1} \equiv 1 (\pmod{p^4})$
WolstenholmePrime.wolstenholme_prime_16483 Two known Wolstenholme primes: 16843 and 2124679
WolstenholmePrime.wolstenholme_bernoulli Equivalently, a prime $p > 7$ is a Wolstenholme prime if it divides the numerator of the Bernoulli number $B_{p-3}$
WolstenholmePrime.wolstenholme_harmonic Another equivalent definition is that a prime $p > 7$ is a Wolstenholme prime if it $p^3$ divides the numerator of the harmonic number $H_{p-1}$
WolstenholmePrime.wolstenholme_prime_infinite It is conjectured that there are infinitely many Wolstenholme primes
WoodallPrimes.infinitely_many_woodall_primes There are infinitely many prime numbers of the form k * 2 ^ k - 1 for k > 1
Conjecture_1_3_to_2_3.conjecture_1_3_to_2_3 Does every finite partially ordered set that is not totally ordered contain two elements
WrittenOnTheWallII.GraphConjecture160.maxTrianglesAtVertex The maximum number of triangles incident to any vertex in $G$
WrittenOnTheWallII.GraphConjecture100.conjecture100 Documented declaration: conjecture100
WrittenOnTheWallII.GraphConjecture101.indepNumDeleteVertex The independence number of the subgraph induced on $V \setminus \{v\}$ (i.e., the graph $G - v$)
WrittenOnTheWallII.GraphConjecture101.alphaCore The $\alpha$-core of $G$: the set of vertices whose removal strictly decreases the independence number
WrittenOnTheWallII.GraphConjecture101.conjecture101 WOWII Conjecture 101 For a simple connected graph
WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample The 11-vertex counterexample: a triangle with four leaves on each of two vertices
WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_connected The counterexample is connected
WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_indepNum The counterexample has independence number nine
WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_bipartiteSize The largest induced bipartite subgraph of the counterexample has ten vertices
WrittenOnTheWallII.GraphConjecture103.wowii103Counterexample_averageEccentricity The counterexample has average eccentricity $30/11$
WrittenOnTheWallII.GraphConjecture103.one_lt_log_thirty_eleven_and_lt_two The logarithm of the counterexample's average eccentricity lies strictly between one and two
WrittenOnTheWallII.GraphConjecture103.conjecture103 WOWII Conjecture 103 For a simple connected graph
WrittenOnTheWallII.GraphConjecture13.conjecture13 Documented declaration: conjecture13
WrittenOnTheWallII.GraphConjecture133.conjecture133 WOWII Conjecture 133: For a simple connected graph
WrittenOnTheWallII.GraphConjecture141.conjecture141 WOWII Conjecture 141 For a simple connected graph
WrittenOnTheWallII.GraphConjecture143.conjecture143 WOWII Conjecture 143: For a simple connected graph
WrittenOnTheWallII.GraphConjecture145.localIndependenceMin localIndependenceMin G is the minimum over all vertices of the local independence number indepNeighborsCard G v
WrittenOnTheWallII.GraphConjecture146.graphSquareRadius The radius of $G^2$ (the graph square): the minimum eccentricity over all vertices of graphSquare G
WrittenOnTheWallII.GraphConjecture146.conjecture146 WOWII Conjecture 146 For a simple connected graph
WrittenOnTheWallII.GraphConjecture16.conjecture16 Documented declaration: conjecture16
WrittenOnTheWallII.GraphConjecture17.conjecture17 Documented declaration: conjecture17
WrittenOnTheWallII.GraphConjecture172.peripheryMaxDegree The maximum original-graph degree among the peripheral vertices
WrittenOnTheWallII.GraphConjecture172.squareMaximumDegreeVertices The maximum-degree vertices of G², presented using finite graph distance
WrittenOnTheWallII.GraphConjecture176.squareMaximumDegreeVertices The maximum-degree vertices of G², presented using finite graph distance
WrittenOnTheWallII.GraphConjecture18.conjecture18 WOWII Conjecture 18 For a simple connected graph
WrittenOnTheWallII.GraphConjecture181.squarePeripheryAverage The average degree, in G², of the maximum-eccentricity vertices of G²
WrittenOnTheWallII.GraphConjecture19.conjecture19 Documented declaration: conjecture19
WrittenOnTheWallII.GraphConjecture198a.conjecture198a WOWII Conjecture 198a For a simple connected graph G, if b(G) ≤ 2 + ecc_avg(G), then G has a Hamiltonian path
WrittenOnTheWallII.GraphConjecture20.conjecture20 Documented declaration: conjecture20
WrittenOnTheWallII.GraphConjecture217.residueEqTwoIndicator The characteristic function for the predicate $\mathrm{residue}\, G = 2$: returns $1$ when $G.\mathrm{residue} = 2$ and $0$ otherwise
WrittenOnTheWallII.GraphConjecture23.conjecture23 Documented declaration: conjecture23
WrittenOnTheWallII.GraphConjecture291.minTrianglesAtVertex The minimum number of triangles incident to any vertex, over all vertices of $G$
WrittenOnTheWallII.GraphConjecture291.freqMinTriangles The number of vertices achieving the minimum triangle count
WrittenOnTheWallII.GraphConjecture291.descDegreeSequence The descending degree sequence of $G$, used as the starting point of the Havel-Hakimi reduction
WrittenOnTheWallII.GraphConjecture291.havelHakimiIterate The Havel-Hakimi sequence of iterates: s i is the result of applying havelHakimiStep $i$ times to the descending degree sequence of $G$
WrittenOnTheWallII.GraphConjecture291.havelHakimiZeroStep The first step $i$ (counting from $0$) at which a zero appears in the Havel-Hakimi reduction of the descending degree sequence of $G$, or in which the sequence has been emptied
WrittenOnTheWallII.GraphConjecture291.conjecture291 WOWII Conjecture 291 For a simple connected graph
WrittenOnTheWallII.GraphConjecture3.conjecture3 Documented declaration: conjecture3
WrittenOnTheWallII.GraphConjecture309.evenHorizontal The number of edges whose endpoints are at the same even distance from $v$
WrittenOnTheWallII.GraphConjecture309.maxEvenCorrection The maximum of $\operatorname{distEven}(v)-\operatorname{evenHorizontal}(v)$
WrittenOnTheWallII.GraphConjecture309.minComplementEdgeNeighborhood The minimum complement-edge neighborhood-union order, when a complement edge exists
WrittenOnTheWallII.GraphConjecture309.conjecture309Statement The universal inequality proposed in WOWII Conjecture 309
WrittenOnTheWallII.GraphConjecture314.largestInducedPathSize The size of a largest induced path of $G$, as a natural number
WrittenOnTheWallII.GraphConjecture314.conjecture314 WOWII Conjecture 314: For every finite simple connected graph $G$ with $n > 1$ vertices, if $G$ is triangle-free and $\mathrm{path}(G) \le 4$, then $G$ is well totally dominated
WrittenOnTheWallII.GraphConjecture32.conjecture32 WOWII Conjecture 32 For a simple connected graph
WrittenOnTheWallII.GraphConjecture322.conjecture322 WOWII Conjecture 322 Let G be a simple connected graph on n ≥ 5 vertices
WrittenOnTheWallII.GraphConjecture33.conjecture33 WOWII Conjecture 33 For a simple connected graph
WrittenOnTheWallII.GraphConjecture34.conjecture34 WOWII Conjecture 34 For a simple connected graph
WrittenOnTheWallII.GraphConjecture36.dp dp G is the number of diametrical pairs of G: the number of unordered pairs {u, v} of vertices at distance diam(G)
WrittenOnTheWallII.GraphConjecture36.wowii36Counterexample A 10-vertex counterexample to Conjecture 36
WrittenOnTheWallII.GraphConjecture36.wowii36Counterexample_connected The counterexample is connected
WrittenOnTheWallII.GraphConjecture36.wowii36Counterexample_radius The counterexample has radius three
WrittenOnTheWallII.GraphConjecture36.wowii36Counterexample_no_inducedPath6 The counterexample contains no induced path on six vertices
WrittenOnTheWallII.GraphConjecture36.wowii36Counterexample_path The largest induced path in the counterexample has exactly five vertices
WrittenOnTheWallII.GraphConjecture36.wowii36Counterexample_dp The counterexample has a unique diametrical pair, namely {2, 3}
WrittenOnTheWallII.GraphConjecture36.conjecture36 WOWII Conjecture 36: For every finite simple connected graph
WrittenOnTheWallII.GraphConjecture4.conjecture4 Documented declaration: conjecture4
WrittenOnTheWallII.GraphConjecture40.conjecture40 Documented declaration: conjecture40
WrittenOnTheWallII.GraphConjecture430a.centerFinset The center of a finite graph
WrittenOnTheWallII.GraphConjecture430a.setNeighborhood DeLaViña's N(S): the union of open vertex neighborhoods
WrittenOnTheWallII.GraphConjecture430a.caroWei The exact rational Caro--Wei sum
WrittenOnTheWallII.GraphConjecture5.conjecture5 WOWII Conjecture 5 For a simple connected graph G, Ls(G) is bounded below by the maximal size of a sphere of radius radius(G) around the centres of G
WrittenOnTheWallII.GraphConjecture6.conjecture6 WOWII Conjecture 6 For a connected graph G we have Ls(G) ≥ 1 + n(G) - m(G) - a(G) where a(G) is defined via independent sets
WrittenOnTheWallII.GraphConjecture61.conjecture61 WOWII Conjecture 61 For a simple connected graph
WrittenOnTheWallII.GraphConjecture65.Counterexample.graph The counterexample: a path on vertices $0,\ldots,12$, with triangles attached at vertices $1$ and $11$
WrittenOnTheWallII.GraphConjecture65.Counterexample.connected The counterexample is connected
WrittenOnTheWallII.GraphConjecture65.Counterexample.min_degree_vertices The minimum-degree vertices of the counterexample are the two path endpoints
WrittenOnTheWallII.GraphConjecture65.Counterexample.max_degree_vertices The maximum-degree vertices of the counterexample are the two triangle attachment points
WrittenOnTheWallII.GraphConjecture65.Counterexample.distMin_min_degree_vertices The minimum distance between minimum-degree vertices is $12$
WrittenOnTheWallII.GraphConjecture65.Counterexample.distMin_max_degree_vertices The minimum distance between maximum-degree vertices is $10$
WrittenOnTheWallII.GraphConjecture65.Counterexample.triangle_not_subset_of_isAcyclic An acyclic induced subgraph cannot contain all three vertices of a triangle
WrittenOnTheWallII.GraphConjecture65.Counterexample.forest_card_le_fifteen Every induced forest in the counterexample has at most $15$ vertices
WrittenOnTheWallII.GraphConjecture65.Counterexample.largest_induced_forest_le_fifteen The largest induced forest in the counterexample has at most $15$ vertices
WrittenOnTheWallII.GraphConjecture65.conjecture65 WOWII Conjecture 65: Does every simple connected graph
WrittenOnTheWallII.GraphConjecture7.conjecture7 WOWII Conjecture 7 For a simple connected graph
WrittenOnTheWallII.Test.HouseGraph House Graph: Square 0-1-2-3-0 with roof 4 connected to 2,3
WrittenOnTheWallII.Test.K4 K4: Complete graph on 4 vertices
WrittenOnTheWallII.Test.PetersenGraph Petersen Graph on 10 vertices
WrittenOnTheWallII.Test.C6 C6: Cycle graph on 6 vertices
WrittenOnTheWallII.Test.Star5 Star5: Star graph with center 0 and 5 leaves
Finset.lcmInterval The least common multiple of ${n+1, \dotsc, n+k}$
GradedMonoid.mk_eq_mk Equality of two elements of GradedMonoid A written with GradedMonoid.mk
GromovPolynomialGrowth.CayleyBall The CayleyBall is the ball of radius n in the Cayley graph of a group G with generating set S
GromovPolynomialGrowth.GrowthFunction The GrowthFunction of a group G with respect to a set S counts the number of group elements that can be reached by words of length at most n in S
GromovPolynomialGrowth.one_mem_cayleyBall The identity is always in the Cayley ball
GromovPolynomialGrowth.cayleyBall_monotone The Cayley ball is monotonic in its radius
GromovPolynomialGrowth.cayleyBall_mul Closure property: if g ∈ CayleyBall S m and h ∈ CayleyBall S n, then g * h ∈ CayleyBall S (m + n)
GromovPolynomialGrowth.cayleyBall_inv If g ∈ CayleyBall S n, then g⁻¹ ∈ CayleyBall S n
GromovPolynomialGrowth.tendsto_atTop_growthFunction_of_infinite In an infinite group, the growth function with respect to a finite generating set is unbounded
GromovPolynomialGrowth.HasPolynomialGrowth A group has polynomial growth if there exists a finite generating set whose growth function is bounded above by a polynomial
GromovPolynomialGrowth.HasSuperPolynomialGrowth Documented declaration: HasSuperPolynomialGrowth
Set.indicatorOne A polymorphic indicator function 𝟙_A which is 1 on A and 0 outside
CategoryTheory.Abelian.Ext.comp_heq Up to HEq, the composition Ext.comp does not depend on the proof of the degree equation
MvPolynomial.poissonBracket Documented declaration: poissonBracket
MvPolynomial.RegularFunction The type of regular functions from $k^σ$ to $k^τ$
MvPolynomial.RegularFunction.Jacobian The Jacobian of a vector valued polynomial function, viewed as a polynomial
MvPolynomial.RegularFunction.comp The composition of two vector valued polynomial functions
MvPolynomial.RegularFunction.aeval The evaluation of a regular function f over k at some point a with coordinates in some algebra over k
MvPolynomial.RegularFunction.comp_aeval aeval is compatible with composition of regular functions
BunyakovskyCondition Bunyakovsky condition A condition on polynomials in the Schinzel and Bunyakovsky conjectures
SchinzelCondition Schinzel condition A condition on sets of polynomials in the Schinzel and Bunyakovsky conjectures
IsSumSqOfLength IsSumSqOfLength n a means that a is a sum of n squares
pythagorasBounds The set of p such that every sum of squares in R is a sum of p squares
IsSumSqOfLength.of_le Padding with zeros: a sum of p squares is a sum of q squares for every q ≥ p
WeierstrassCurve.realPeriodIntegrand Documented declaration: realPeriodIntegrand
WeierstrassCurve.realPeriod Documented declaration: realPeriod
WeierstrassCurve.complexPeriodIntegrand The density of $\Omega_{\mathbb{C}}$ on the $x$-line: $1 / |\Psi_2^2(x)|$
WeierstrassCurve.complexPeriod The complex period as an integral: $\Omega_{\mathbb{C}}(E) = \int_{E(\mathbb{C})} |\omega \wedge \bar\omega| = 4 \int_{\mathbb{C}} dA(x) / |\Psi_2^2(x)|$
WeierstrassCurve.minimalDiscriminantExponent Documented declaration: minimalDiscriminantExponent
WeierstrassCurve.minimalDiscriminant Documented declaration: minimalDiscriminant
WeierstrassCurve.Affine.Point.naiveHeight The logarithmic height hₓ of the x-coordinate with value zero at infinity
WeierstrassCurve.Affine.Point.canonicalHeight The canonical height lim_n 4⁻ⁿhₓ(2ⁿP) in terms of the logarithmic height hₓ
WeierstrassCurve.Affine.Point.neronTatePairing The Néron--Tate height pairing, defined by polarisation of the canonical height
WeierstrassCurve.Affine.Point.neronTateMatrix The Gram matrix of the Néron--Tate height pairing on any family of K-rational points
WeierstrassCurve.Affine.Point.regulator The absolute determinant of the height pairing on a finite basis of E(K) modulo torsion
WeierstrassCurve.Projective.nonsingularReduction The points with an integral projective representative whose reduction is non-singular
WeierstrassCurve.Projective.tamagawaNumber The local Tamagawa number attached to an integral model
WeierstrassCurve.Projective.tamagawaProduct The product of local Tamagawa numbers for minimal integral models at all the non-archimedean completions
WeierstrassCurve.Affine.galoisRepresentation The natural Galois action on geometric non-singular points with their discrete topology
WeierstrassCurve.Affine.tateShafarevich The subgroup of the first continuous Galois cohomology group consisting of classes whose restrictions vanish at every infinite and finite place
ProjectiveSpace The projective space over a scheme S, with homogeneous coordinates indexed by n
SheafOfModules.VectorBundleData A vector bundle is a sheaf of modules that is locally isomorphic to a free sheaf
SheafOfModules.IsVectorBundleWithRank A class for vector bundles of constant finite rank
IsEquidistributed Documented declaration: IsEquidistributed
IsEquidistributedModuloOne Documented declaration: IsEquidistributedModuloOne
translateSet The translate of a set Ω by a vector t
exponentialCharacter The exponential character x ↦ exp(2π i ⟪ξ, x⟫)
continuous_exponentialCharacter The exponential character is continuous
norm_exponentialCharacter The exponential character has norm one
exponentialCharacter_memLp On a finite-measure set, every exponential character belongs to L^2
spectralPair Documented declaration: spectralPair
isSpectral A set is spectral if it belongs to a spectral pair
tilesByTranslation A set Ω tiles ℝ^d by translation if there exists a set T of translation vectors such that the translates of Ω by T cover ℝ^d without overlap
HasFabryGaps A sequence of natural numbers n₀ < n₁ < ... is said to have Fabry gaps if nₖ / k → ∞
HasFejerGaps A sequence of natural numbers n₀ < n₁ < ... is said to have Fejér gaps if ∑' 1 / nₖ < ∞
additiveChar The additive character e(x) = e ^ (2 * π * i * x), a shorthand used across several exponential-sum problems
Real.iteratedLog The iterated logarithm of x is the number of times the natural logarithm must be iteratively applied before the result is less than or equal to 1
Nat.iteratedLog The iterated logarithm of a natural number in base b: the number of times Nat.log b must be iteratively applied to n before the result is at most 1
Set.IsAPOfLengthWith Documented declaration: IsAPOfLengthWith
List.IsAPOfLengthWith A list version of Set.IsAPOfLengthWith
Set.IsAPOfLengthWith.zero An arithmetic progression with first term a and difference d is of length zero if and only if s is empty
Set.IsAPOfLengthWith.one An arithmetic progression with first term a and difference d is of length one if and only if s is a singleton
List.IsAPOfLengthWith.zero An arithmetic progression with first term a and difference d is of length zero if and only if s is empty
List.IsAPOfLengthWith.one An arithmetic progression with first term a and difference d is of length one if and only if s is a singleton
Set.isAPOfLengthWith_pair In an abelian additive group α, the set {a, b} with a ≠ b is an arithmetic progression of length 2 with first term a and difference b - a
Nat.isAPOfLengthWith_pair The set {a, b} : Set ℕ with a < b is an arithmetic progression of length 2 with first term a and difference b - a
Set.IsAPOfLength The predicate that a set s is an arithmetic progression of length l (possibly infinite)
List.IsAPOfLength A list version of Set.IsAPOfLength
Set.IsAPOfLength.zero Only the empty set is a finite arithmetic progression of length $0$
Set.IsAPOfLength.one Only singletons are finite arithmetic progressions of length $1$
Set.IsAPOfLength.congr If a set is an arithmetic progression of lengths l₁ and l₂, then the lengths are equal
List.IsAPOfLength.zero Only the empty list is a finite arithmetic progression of length $0$
List.IsAPOfLength.one Only singletons are finite arithmetic progressions of length $1$
List.IsAPOfLength.congr If a list is an arithmetic progression of lengths l₁ and l₂, then the lengths are equal
Set.not_isAPOfLength_empty The empty set is not an arithmetic progression of positive length
Set.IsAPOfLengthFree We say that a set s is free of arithmetic progressions of length l if s contains no non-trivial arithmetic progressions of length l
Set.isAPOfLengthFree_one Any set is free of arithmetic progressions of length 1, because such APs are all trivial
Set.isAPOfLengthFree_zero Any set is free of arithmetic progressions of length 0, because such APs are all trivial
Set.IsAPOfLength.not_isAPOfLengthFree Any non-trivial arithmetic progression cannot be free of arithmetic progressions
Set.IsAPOfLengthFree.maxCard Define the largest possible size of a subset of $\{1, \dots, N\}$ that does not contain any non-trivial $k$-term arithmetic progression
Set.ContainsAP A set A contains an arithmetic progression of length k with difference d
HasMonotoneAP A function f : β → α has a monotone k-term arithmetic progression if there exists a choice of indices `b 1 < b 2 <
Finset.maxAPFreeCard Define the largest possible size of a subset of a finset s that does not contain any non-trivial k-term arithmetic progression
Set.IsMulBasis An additive basis of some order
Set.not_isMulBasisOfOrder_zero No set is an additive basis of order 0
Set.IsAsymptoticMulBasis An asymptotic additive basis of some order
Set.IsWeakMulBasis A weak additive basis of some order
AdditiveCombinatorics.sumConv The sum convolution of two functions f, g : ℕ → R, also known as the Cauchy product
AdditiveCombinatorics.sumRep The number of sum representations is the sum convolution of A's indicator function with itself: $1_A\ast 1_A(n)$
maxCosetDim_univ The maximum coset dimension in the entire vector space $V$ is exactly the dimension of $V$
maxCosetDim_empty The maximum coset dimension in the empty set $\emptyset$ is $0$
maxCosetDim_mono If $A \subseteq B$, then the maximum coset dimension achievable in $A$ cannot exceed the maximum coset dimension achievable in $B$
maxCosetDim_affineSubspace If the target set $A$ is already an affine subspace, the function returns exactly the rank of its direction
maxCosetDim_singleton Similar to the empty set, a set containing exactly one vector should yield a maximum dimension of 0
maxCosetDim_vadd Translating a set by a vector does not change its maximum coset dimension
Finset.IsRatioBasis.mono_left Shrinking the target set preserves being a difference basis
Finset.IsRatioBasis.mono_right Enlarging the basis preserves being a difference basis
Finset.IsRatioBasis.card_le_sq A difference basis for $S$ has at least $\sqrt{|S|}$ elements: $|S| \le |A|^2$
Finset.restrictedSumset The restricted sumset of a set $S$, denoted $S \hat{+} S$, is the set $\lbrace s_1 + s_2 : s_1, s_2 \in S, s_1 \neq s_2 \rbrace$
allUniqueSums allUniqueSums A is the set of elements in α that can be written as the sum of exactly one unordered pair of elements from A
HasNoUniqueRepresentation Documented declaration: HasNoUniqueRepresentation
IsSyndetic A set $A$ of natural numbers is said to have bounded gaps if there exists an integer $p$ such that $A ∩ [n, n + 1, ..., n + p]$ is nonempty for all $n$
IsSidon A Sidon set is a set, such that such that all pairwise sums of elements are distinct apart from coincidences forced by the commutativity of addition
Set.IsMaximalSidonSetIn IsMaximalSidonSetIn A N means A ⊆ {1, …, N} is Sidon and cannot be extended within {1, …, N} while remaining Sidon
Set.IsMaximalSidonSetIn.subset If A is a maximal Sidon set in {1, …, N}, then A ⊆ {1, …, N}
Set.IsMaximalSidonSetIn.isSidon If A is a maximal Sidon set in {1, …, N}, then A is Sidon
Set.IsMaximalSidonSetIn.maximal Maximality condition unpacked
Finset.maxSidonSubsetCard The maximum size of a Sidon set in the supplied Finset
Finset.IsSidon.insert_ge_max' If A is finite Sidon, then A ∪ {s} is also Sidon provided s ≥ A.max + 1
Finset.greedySidon.go Given a finite Sidon set A and a lower bound m, go finds the smallest number m' ≥ m such that A ∪ {m'} is Sidon
Finset.greedySidon.aux Main search loop for generating the greedy Sidon sequence
Finset.greedySidon greedySidon is the sequence obtained by the initial set $\{1\}$ and iteratively obtaining the next smallest integer that preserves the Sidon property of the set
Finset.greedySidonBelow The greedy Sidon set in {1, …, N}: starting from ∅, iterate through 1, …, N and include x if and only if A ∪ {x} remains Sidon
Digraph.IsTournament A tournament is a directed graph in which every pair of distinct vertices is connected by exactly one directed edge
Digraph.IsOrientation An orientation of an undirected simple graph H is a digraph D whose edges are obtained by directing each edge of H in exactly one direction
Digraph.Hom The type of digraph homomorphisms D →g D'
Digraph.homCount The number of homomorphisms from a digraph D to a digraph D'
Digraph.homDensity The homomorphism density $t_D(D')$ of a digraph $D$ in a digraph $D'$
Hypergraph.IsUniform Every edge has exactly k vertices
Hypergraph.IsProperColoring A weak proper coloring gives two vertices different colors in every edge
Hypergraph.IsNColorable A weak proper coloring using at most n colors
Hypergraph.ofEdgeFamily A family of finite edges on an explicit vertex set, retaining isolated vertices
Hypergraph.IsUniform.finite_edge Every edge of a uniform hypergraph is finite
Hypergraph.isProperColoring_of_injective An injective coloring is proper when every edge has at least two vertices
Hypergraph.IsUniform.isProperColoring_of_injective An injective coloring is proper when uniformity is at least two
Hypergraph.fixedSizeFamilies All r-uniform hypergraphs on n vertices with exactly m edges
Hypergraph.matchingProbability Probability of a matching in a uniformly chosen fixed-size edge family
Hypergraph.linearIndependenceNumber Smallest independence number of a linear three-uniform hypergraph on n vertices
Hypergraph.localTransversalBound Maximum global transversal size under a local vertex-set constraint
Hypergraph.subfamilyTransversalBound Global transversal bound for a uniform family whose small subfamilies have small transversals
Hypergraph.minIntersectingEdges Least number of edges in an intersecting r-uniform family with transversal size r
Hypergraph.sparseConfigurationThreshold Least vertex count forcing the Brown--Erdős--Sós little-o conclusion
Hypergraph.edgeDensity Edge density normalized by the number of possible uniform edges
Hypergraph.HasDensityJump A density increase on growing induced vertex sets for every growing host sequence
Hypergraph.balancedColoringThreshold The least size above which a coloring gives both colors a positive share of every induced complete uniform hypergraph
Set.HasFiniteTransversal A family of finite sets has a transversal with at most k vertices
Finset.IsUniform Every edge has exactly r vertices
Finset.toHypergraph The hypergraph with edge family H and ground vertex set S
Finset.isUniform_toHypergraph_iff Conversion preserves uniformity
Finset.IsProperHypergraphColoring A weak proper coloring has no monochromatic edge
Finset.isProperColoring_toHypergraph_iff Conversion preserves weak proper colorings
Finset.HypergraphColorable The edge family admits a weak proper coloring with k colors
Finset.hypergraphColorable_two_iff Two-colorability agrees with the existing Property B predicate
Finset.HasHypergraphChromaticNumber The weak chromatic number is exactly k
Finset.IsHypergraphIndependent No edge is contained in this set of vertices
Finset.hypergraphDegree The number of hyperedges containing a vertex
Finset.IsLinearHypergraph Distinct edges intersect in at most one vertex
Finset.ContainsSubgraph Some m vertices span at least k edges
Finset.hypergraphInduce The edges induced on a vertex set
Finset.ContainsCompleteHypergraph A copy of the complete r-uniform hypergraph on k vertices
Finset.HasHypergraphMatching A collection of k pairwise vertex-disjoint edges
Finset.IsBlockDesign Every t-set of ground vertices lies in exactly one block
Finset.IsPairwiseBalancedDesign A pairwise balanced design, permitting a block equal to the ground set
Finset.blockSizeProfile The nondecreasing block-size sequence, retaining repeated sizes
Finset.IsCompleteHypergraphDecomposition Partition into complete r-graphs with either r or r+1 vertices
Finset.ContainsCompletePartiteHypergraph A complete t-partite t-uniform hypergraph with r vertices in each part
Finset.HasRepeatedDisjointUnion Two distinct disjoint pairs of edges with the same union
Finset.containsSubgraph_iff The cardinal bound in ContainsSubgraph counts edges, not copies with labels
Finset.not_colorable_singleton A singleton edge prevents any weak proper coloring
Finset.hypergraphIndependenceNumber The largest cardinality of an independent vertex set in a finite ground type
Finset.propertyBWitnesses Subsets of the incident vertices witnessing Property B
Hypergraph.extremalNumber Maximum number of edges among r-uniform families satisfying P on n labeled vertices
Hypergraph.cliqueExtremalNumber The extremal number forbidding a complete uniform hypergraph
Hypergraph.configurationExtremalNumber The extremal number forbidding e edges on v vertices (not necessarily induced)
Hypergraph.partiteExtremalNumber The extremal number forbidding a complete balanced multipartite uniform hypergraph
Hypergraph.minChromaticEdges Minimum edge count for chromatic number k, with infinity when no such graph exists
Hypergraph.blockSizeProfiles All block-size sequences of pairwise balanced designs on n labeled vertices
UniformHypergraph A uniform Mathlib hypergraph whose vertex set is the entire ambient type
UniformHypergraph.ofFinset A finite uniform edge family viewed as a uniform hypergraph
UniformHypergraph.IsCompleteOn Every k-element subset of S is an edge
UniformHypergraph.cliqueSizes The sizes of the finite maximal complete subgraphs
UniformHypergraph.IsProperColoring Weak proper coloring of the underlying Mathlib hypergraph
UniformHypergraph.isProperColoring_of_injective An injective coloring is proper when every edge has at least two vertices
UniformHypergraph.isProperColoring_ofFinset_iff The finite and possibly infinite presentations have the same proper colorings
UniformHypergraph.chromaticCardinal The infimum of cardinalities of color types admitting a proper coloring
UniformHypergraph.chromaticCardinal_le A proper coloring bounds the chromatic cardinal by the number of colors
UniformHypergraph.chromaticCardinal_le_mk The vertex type itself always supplies enough colors when 2 ≤ k
UniformHypergraph.Appears An injective vertex map carrying every edge of F to an edge of H
UniformHypergraph.Appears.trans Appearance is transitive under composition of injective vertex maps
UniformHypergraph.IsNColorable A weak proper coloring using at most n colors
UniformHypergraph.HasAvoidingChromaticCardinal Some hypergraph of the same uniformity and chromatic cardinal κ omits F
UniformHypergraph.IsObligatory A finite uniform hypergraph is obligatory if it appears in every hypergraph of that uniformity whose chromatic cardinal exceeds ℵ₀
LatinSquare A latin square of order n is an n × n matrix with entries in Fin n such that each symbol appears exactly once in every row and exactly once in every column
IsTransversal Documented declaration: IsTransversal
IsNearTransversal Documented declaration: IsNearTransversal
numTransversals The number of transversals of a latin square L, defined as the cardinality of the type of column-selectors σ : Fin n → Fin n satisfying IsTransversal L σ
LimitObjects.UnitInterval The unit interval [0, 1]
LimitObjects.Graphon A graphon is a symmetric, measurable function W : [0, 1] × [0, 1] → [0, 1]
LimitObjects.Graphon.ext Extensionality principle for Graphon
LimitObjects.Graphon.evalSym2 Evaluate a graphon on an unordered pair of vertices in Sym2 UnitInterval as a real number
LimitObjects.Graphon.evalEdge Given a configuration x : V → UnitInterval and an edge e : Sym2 V, evaluate W on the endpoints as a real number
LimitObjects.graphonHomIntegrand The integrand for the homomorphism density $t(H, W)$ of a simple graph $H$ in a graphon $W$ given a configuration x : V → [0, 1]
LimitObjects.graphonHomDensity The homomorphism density $t(H, W)$ of a finite simple graph $H$ in a graphon $W$ on $[0, 1]$ equipped with the Lebesgue measure volume
LimitObjects.graphonEdgeDensity The edge density of a graphon $W$ on $[0, 1]^2 \to [0, 1]$ equipped with the Lebesgue measure volume
LimitObjects.Tournamenton A tournamenton is a measurable function W : [0, 1] × [0, 1] → [0, 1] satisfying the tournament condition W(x, y) + W(y, x) = 1
LimitObjects.Tournamenton.ext Extensionality principle for Tournamenton
LimitObjects.digraphEdgeFinset The directed edge finset of a finite digraph
LimitObjects.tournamentonHomIntegrand The integrand for the homomorphism density $t_{\vec{T}}(W)$ of a digraph $D$ in a tournamenton $W$ given a configuration x : V → [0, 1]
LimitObjects.tournamentonHomDensity The homomorphism density $t_{\vec{T}}(W)$ of a digraph $D$ in a tournamenton $W$ on $[0, 1]$ equipped with the Lebesgue measure volume
Combinatorics.hypergraphRamsey The r-uniform hypergraph Ramsey number R_r(n)
Combinatorics.le_hypergraphRamsey n ≤ hypergraphRamsey r n when the set is nonempty
Combinatorics.hypergraphRamsey_self hypergraphRamsey r r = r for any r
Combinatorics.Diagonal.HasRamseyProperty Documented declaration: HasRamseyProperty
Combinatorics.Diagonal.HasRamseyProperty.mono Monotonicity
Combinatorics.Diagonal.hasRamseyProperty_zero_left Base case s = 0
Combinatorics.Diagonal.hasRamseyProperty_zero_right Base case t = 0
Combinatorics.Diagonal.forall_two_subset_singleton A singleton has no 2-element subsets, hence is monochromatic of any colour
Combinatorics.Diagonal.hasRamseyProperty_one_left Base case s = 1
Combinatorics.Diagonal.hasRamseyProperty_one_right Base case t = 1
Combinatorics.Diagonal.card_false_true_split Pigeonhole step: splitting V \ {v} by colour
Combinatorics.Diagonal.mono_insert Extending a b-monochromatic clique S by a vertex v that is b-adjacent to every element of S
Combinatorics.Diagonal.HasRamseyProperty.step Recurrence
Combinatorics.Diagonal.hasRamseyProperty_choose Binomial bound via Erdős–Szekeres induction
Combinatorics._root_.Combinatorics.hypergraphRamsey_two_le_four_pow Erdős–Szekeres 1935 in hypergraphRamsey form: Combinatorics.hypergraphRamsey 2 k ≤ 4 ^ k for all k
Finset.HasPropertyB A family F of finite sets has property B (equivalently, chromatic number 2) if there is a 2-colouring of the ground set under which no member of F is monochromatic
IsSunflowerWithKernel A sunflower F with kernel S is a collection of sets in which all possible distinct pairs of sets share the same intersection S
IsSunflower A sunflower F is a collection of sets in which all possible distinct pairs of sets share the same intersection
Finset.UnionFree A family F of finsets is union-free if there are no solutions to A ∪ B = C with A, B, C distinct members of F
Finset.SubfamilyUnionFree A family F of finsets is subfamily union-free if no member is the union of a nonempty subfamily of the others
Shatters A set family 𝒜 shatters a set A if restricting 𝒜 to A gives rise to all subsets of A
HasVCDimAtMost Documented declaration: HasVCDimAtMost
Combinatorics.cardinalPartitionRel cardinalPartitionRel μ r γ ν asserts the multicolor
SimpleGraph.annihilationNumber The annihilation number of a graph
SimpleGraph.annihilationNumber' Computes the annihilation number of a graph G
SimpleGraph.IsBalanced We call a graph $D$-balanced (or $D$-almost-regular) if the maximum degree is at most $D$ times the minimum degree
SimpleGraph.IsChordal A graph is chordal if it contains no induced cycle of length k ≥ 4
SimpleGraph.IsCliqueEdgePartition Documented declaration: IsCliqueEdgePartition
SimpleGraph.IsSplitGraph A graph is a split graph if its vertex set splits into a clique and an independent set
SimpleGraph.exists_isCliqueEdgePartition_card_eq Every finite graph has a clique edge partition consisting of its edges
SimpleGraph.IsSplitGraph.isChordal Every split graph is chordal
SimpleGraph.isCliqueEdgePartition_bot_empty The empty family partitions the edges of an edgeless graph
SimpleGraph.isSplitGraph_bot An edgeless graph is split, including when its vertex type is empty
SimpleGraph.cycleLengths G.cycleLengths is the set of lengths of the cycles in G
SimpleGraph.oddCycleLengths G.oddCycleLengths is the set of lengths of odd cycles in G
SimpleGraph.circumference circumference G is the length of the longest cycle in G
SimpleGraph.cliqueSizes The set of sizes of the cliques (maximal complete subgraphs) of G
SimpleGraph.trianglesContaining The triangles of G containing the edge uv, that is, the 3-cliques of G whose vertex set contains both endpoints of uv
SimpleGraph.IsCriticalEdges A set of edges is critical if deleting them reduces the chromatic number
SimpleGraph.IsCriticalEdge An edge is critical if deleting it reduces the chromatic number
SimpleGraph.Subgraph.IsCriticalVerts A set of vertices is critical if deleting them reduces the chromatic number
SimpleGraph.Subgraph.IsCriticalVertex A vertex is critical if deleting it reduces the chromatic number
SimpleGraph.IsCritical A graph G is k-critical (or vertex-critical) if its chromatic number is k, and deleting any single vertex reduces the chromatic number
SimpleGraph.Cocolorable G.Cocolorable n means that the vertices of G can be colored with n colors so that each color class induces either an independent set or a complete graph
SimpleGraph.cochromaticNumber The cochromatic number of a graph is the minimal number of colors needed to color its vertices so that each color class induces either an independent set or a complete graph
SimpleGraph.chromaticCardinal The chromatic cardinal is the minimal number of colors need to color it
SimpleGraph.indepNumK The maximum size of the union of k finite independent sets
SimpleGraph.CDSColorable Documented declaration: CDSColorable
SimpleGraph.IsRainbow A homomorphism f : H →g G is rainbow for an edge labeling c of G if it maps distinct edges of H to edges of G with distinct labels
SimpleGraph.antiRamseyNum The anti-Ramsey number
SimpleGraph.card_edgeFinset_completeGraph The number of edges of the complete graph on a finite vertex type V is #V choose 2
SimpleGraph.card_edgeFinset_completeGraph_fin Specialisation of card_edgeFinset_completeGraph to Fin n
SimpleGraph.InternallyDisjoint Two walks are internally disjoint if they share no vertices other than their endpoints
SimpleGraph.InfinitelyConnected We say a graph is infinitely connected if any two vertices are connected by infinitely many pairwise disjoint paths
SimpleGraph.IsKConnected G is k-connected: it has more than k vertices, and it stays connected after the removal of any set of fewer than k vertices
SimpleGraph.not_isKConnected_of_card_le A graph on at most k vertices is not k-connected
SimpleGraph.bfsStep One round of breadth-first search: G.bfsStep s consists of the vertices of s together with their neighbours
SimpleGraph.bfsIterate Iterate G.bfsStep at most n times, stopping as soon as no new vertex shows up
SimpleGraph.reachableFinset The finset of vertices reachable from u, computed by breadth-first search
SimpleGraph.decidableReachable Decides reachability of vertices u and v by performing a breadth-first search from u
SimpleGraph.decidablePreconnected Decides preconnectedness of G by checking whether the vertex set is empty and, if not, by performing a breadth-first search from an arbitrarily chosen vertex
SimpleGraph.decidableConnected Decides preconnectedness of G by checking whether the vertex set is empty and, if not, by performing a breadth-first search from an arbitrarily chosen vertex
SimpleGraph.cvetkovic Documented declaration: cvetkovic
SimpleGraph.Cycle A cycle of G, carrying its basepoint
SimpleGraph.Cycle.edges The edges a cycle traverses
SimpleGraph.Cycle.length The length of a cycle, its number of edges
SimpleGraph.IsBridgeless G is bridgeless if none of its edges is a bridge
SimpleGraph.edgeFinset_eq_empty_of_isBridgeless_of_isAcyclic In a forest every edge is a bridge, so an acyclic bridgeless graph has no edges at all
SimpleGraph.cycleRank The cycle rank of G (or cyclomatic number) is the minimum number of edges that must be removed to eliminate all cycles
SimpleGraph.IsDecomposition D is a decomposition of G into edge-disjoint subgraphs: the edge sets of the members of D are pairwise disjoint and their union is the edge set of G
SimpleGraph.neighborsWithin The neighbours of v lying inside s
SimpleGraph.IsDegenerate H is r-degenerate when every induced subgraph has a vertex of degree at most r, that is, every nonempty vertex set contains a vertex with at most r neighbours inside it
SimpleGraph.averageDegree The average degree of G
SimpleGraph.degreeMultiset The multiset of degrees of a graph
SimpleGraph.degreeSequence The degree sequence of a graph, sorted in nondecreasing order
SimpleGraph.degreeSequenceMultiplicity The maximum number of occurrences of any term of the degree sequence of G
SimpleGraph.non_edge_neighborhood_card Cardinality of the union of the neighbourhoods of the ends of the non-edge e
SimpleGraph.NG Minimum size of the neighbourhood of a non-edge of G
SimpleGraph.secondSmallestDegree Documented declaration: secondSmallestDegree
SimpleGraph.exists_distinct_degree_one_of_secondSmallestDegree_eq_one In a nontrivial preconnected finite graph, second-smallest degree one forces two leaves
SimpleGraph.numTrianglesAtVertex The number of triangles (3-cliques) of G incident to vertex v: the number of 3-element cliques containing v
SimpleGraph.degreeL2Norm The length of a graph: the square root of the sum of the squares of degrees
SimpleGraph.countDegreeK The number of vertices of degree k in G
SimpleGraph.diam_eq_zero_of_subsingleton The diameter is the greatest distance between any two vertices
SimpleGraph.IsDominating A set D is a dominating set for G if every vertex of G is either in D or adjacent to a vertex of D
SimpleGraph.IsNDominatingSet An n-dominating set is a dominating set with n vertices
SimpleGraph.dominationNumber The domination number of a graph G is the minimum size of a dominating set
SimpleGraph.computable_dom_num Computable domination number via powerset enumeration
SimpleGraph.IsTotalDominating A set D is a total dominating set if every vertex is adjacent to a vertex in D
SimpleGraph.IsTotalNDominatingSet An n-total dominating set is a total dominating set with n vertices
SimpleGraph.totalDominationNumber The total domination number of G
SimpleGraph.IsConnectedDominating A set is a connected dominating set if it is dominating and induces a connected subgraph
SimpleGraph.connectedDominationNumber The connected domination number of G
SimpleGraph.IsEdgeCover A set of edges is an edge cover if every vertex is incident to some edge in it
SimpleGraph.edgeCoverNumber The minimum edge cover number of G
SimpleGraph.exists_isNDominatingSet_dominationNumber The set Finset.univ is a dominating set of any graph, so the set of sizes of dominating sets is nonempty and dominationNumber is attained by some dominating set
SimpleGraph.dominationNumber_le_of_isDominating If D is a dominating set of G with n elements then γ(G) ≤ n
SimpleGraph.dominationNumber_le_dominationNumber_boxProd Projection bound: γ(G) ≤ γ(G □ H) whenever H has a vertex
SimpleGraph.maxEccentricityVertices The set of vertices of maximum eccentricity
SimpleGraph.averageEccentricity The average eccentricity of a graph G: the mean of G.eccent v over all vertices, converted to a real number
SimpleGraph.computable_ediam The diameter of a finite graph as a computable natural number: the maximum BFS distance (computable_dist) over all ordered pairs of vertices
SimpleGraph.ediam_eq_computable For a connected finite graph, the extended diameter ediam equals the computable diameter
SimpleGraph.computable_eccent The eccentricity of a vertex u in a finite graph as a computable natural number: the maximum BFS distance (computable_dist) from u to any vertex
SimpleGraph.computable_radius The radius of a finite graph as a computable natural number: the minimum, over all vertices, of the computable eccentricity
SimpleGraph.eccent_eq_computable For a connected finite graph, the eccentricity eccent u equals the computable eccentricity
SimpleGraph.radius_eq_computable For a connected finite graph, the radius radius equals the computable radius
SimpleGraph.IsEdgeColouring IsEdgeColouring H c asserts that c : ι → SimpleGraph V is a partition of the edges of H: the colour classes cover H (H = ⨆ i, c i) and are pairwise disjoint
SimpleGraph.TopEdgeLabeling.CliqueFree A labelling of the edges of the complete graph on V by ι is n-clique-free when no colour class contains an n-clique, that is, when it admits no monochromatic n-clique
SimpleGraph.fractionalAlpha Fractional alpha
SimpleGraph.harmonicIndex The harmonic index of G, defined as ∑_{uv ∈ E(G)} 2 / (deg u + deg v), where deg is the degree of a vertex
SimpleGraph.homCount The number of graph homomorphisms H → G, as a natural number
SimpleGraph.homDensity The homomorphism density t(H, G) of a simple graph H in a simple graph G with finite vertex sets is #Hom(H, G) / |V(G)|^{|V(H)|}
SimpleGraph.homDensity_le_one The homomorphism density t(H, G) is at most 1
SimpleGraph.hypercube hypercube n is the n-dimensional hypercube graph Qₙ: the vertices are the n-bit vectors, and two vertices are adjacent when they differ in exactly one coordinate
SimpleGraph.aprime The maximum cardinality among all independent sets s that maximize the quantity |s| - |N(s)|, where N(s) is the neighborhood of the set s
SimpleGraph.indepNeighborsCard Independence number of the neighbourhood of v
SimpleGraph.indepNeighbors The same quantity as a real number
SimpleGraph.averageIndepNeighbors Average of indepNeighbors over all vertices
SimpleGraph.l Abbreviation for the average independence number of the neighborhoods
SimpleGraph.l_avg The same quantity under a different name, used in some conjectures
SimpleGraph.computable_indep_num Computable independence number via powerset enumeration
SimpleGraph.indep_num_eq_computable Equivalence between computable and noncomputable independence number
SimpleGraph.maxLocalIndependence Maximum local independence number over all vertices
SimpleGraph.minLocalIndependence Minimum local independence number over all vertices
SimpleGraph.internalEdges Edges of G whose two endpoints lie in S
SimpleGraph.IsTwoIndependent A vertex set is 2-independent when its induced graph has maximum degree at most one
SimpleGraph.indepNumOn Maximum cardinality of an independent subset contained in B
SimpleGraph.alphaTwo Maximum cardinality of a 2-independent vertex set
SimpleGraph.lowDegreeLayer The set of vertices of degree at most two
SimpleGraph.largestInducedForestSize largestInducedForestSize G is the size of a largest induced forest of G
SimpleGraph.largestInducedBipartiteSubgraphSize largestInducedBipartiteSubgraphSize G is the size of a largest induced bipartite subgraph of G
SimpleGraph.induce_isBipartite_iff_exists_coloring An induced subgraph is bipartite exactly when its vertices admit a two-coloring
SimpleGraph.computableLargestInducedBipartiteSubgraphSize Computable largest induced bipartite subgraph size via powerset enumeration
SimpleGraph.largestInducedBipartiteSubgraphSize_eq_computable The computable largest induced bipartite subgraph size agrees with the sSup definition
SimpleGraph.b b G is the number of vertices of a largest induced bipartite subgraph of G
SimpleGraph.path_start_unique_edge In a path, at most one edge is incident to the first vertex
SimpleGraph.path_incident_edges_le_two In a path, at most two edges are incident to any fixed vertex w: among any three vertices joined to w by edges of the path, two coincide
SimpleGraph.exists_subpath_between A path visiting a and b contains a subpath between a and b (in one of the two directions) whose support and edges lie in the original path
SimpleGraph.tree_isPath_eq_cons_nil In a tree, the unique path between two adjacent vertices is the single edge
SimpleGraph.tree_center_mem_support Documented declaration: tree_center_mem_support
SimpleGraph.tree_edge_mem_of_center_mem Documented declaration: tree_edge_mem_of_center_mem
SimpleGraph.IsTree.girth_le_length_add_two_of_two_adj Documented declaration: girth_le_length_add_two_of_two_adj
SimpleGraph.isIndepSet_neighborSet_of_four_le_girth Triangle-free bridge: in a graph of girth at least 4, the neighborhood of any vertex is an independent set
SimpleGraph.IsTree.card_support_inter_closedNeighborFinset_le_three Overlap lemma: on a path p in an induced tree on s, at most 3 vertices lie in the closed neighborhood of a fixed vertex v ∈ s
SimpleGraph.exists_indepSet_finset_neighbors Extract an explicit independent neighbour set witnessing indepNeighborsCard
SimpleGraph.isTree_induce_insert_indepSet_neighbors A vertex together with an independent set of its neighbours induces a star, hence a tree
SimpleGraph.indepNeighborsCard_add_one_le_largestInducedTreeSize The star bound: indepNeighborsCard G v + 1 ≤ largestInducedTreeSize G
SimpleGraph.indepNeighborsCard_le_degree The independence number of a neighbourhood is at most the degree
SimpleGraph.maxDegree_add_girth_le_largestInducedTreeSize_add_three Main counting theorem: in a connected graph with a cycle and girth at least 4, Δ(G) + girth(G) ≤ largestInducedTreeSize G + 3
SimpleGraph.johnson The Johnson graph $J(n, k)$ has as vertices the $k$-subsets of an $n$-set
SimpleGraph.largestInducedTreeSize largestInducedTreeSize G is the number of vertices in a largest induced subtree of G
SimpleGraph.card_le_largestInducedTreeSize The order of any induced tree is bounded by largestInducedTreeSize
SimpleGraph.one_le_largestInducedTreeSize Every finite graph with a vertex has a nonempty induced tree
SimpleGraph.IsTree.induce_insert_of_unique_adj Attaching a new vertex along its unique neighbor in an induced tree gives a larger induced tree
SimpleGraph.Walk.induce_support_isTree_of_length_eq_dist A shortest walk induces a tree on its support
SimpleGraph.girth_sub_one_le_largestInducedTreeSize Removing one vertex from a shortest cycle gives an induced tree
SimpleGraph.Connected.exists_induced_tree_containing_pair Every pair of vertices in a connected graph lies in an induced tree
SimpleGraph.exists_maximum_induced_tree_containing A nonempty family of induced trees with prescribed vertices has a largest member
SimpleGraph.Connected.exists_adj_finset_compl A nonempty proper vertex set in a connected graph has an edge across its boundary
SimpleGraph.exists_two_adj_of_maximum_induced_tree_containing A boundary vertex of a largest prescribed induced tree has two neighbors in the tree
SimpleGraph.IsTree.girth_add_one_le_card_of_two_leaves_of_two_adj Two leaves outside the boundary cycle force an extra two vertices in the induced tree
SimpleGraph.girth_add_one_le_largestInducedTreeSize_of_two_leaves Two distinct leaves in a connected cyclic graph yield an induced tree of order at least girth plus one
SimpleGraph.lovaszThetaFunction Lovász Theta Function ($\vartheta(G)$)
SimpleGraph.lovaszThetaFunction_isEmpty The Lovász theta function of a graph on an empty vertex type is $0$
SimpleGraph.one_le_lovaszThetaFunction The Lovász theta function of any nonempty graph is at least $1$
SimpleGraph.lovaszThetaFunction_anti Adding edges to a graph can only decrease or preserve its Lovász theta function
SimpleGraph.lovaszThetaFunction_top The Lovász theta function of the complete graph $K_n$ ($n \ge 1$) is $1$
SimpleGraph.lovaszThetaFunction_bot The Lovász theta function of the edgeless graph $\overline{K_n}$ is $n$
SimpleGraph.lovaszThetaFunction_le_card The Lovász theta function of any graph is at most its number of vertices
SimpleGraph.matchingNumber matchingNumber G is the size of a maximum matching of G
SimpleGraph.IsEdgeMatching A finite set of edges M is a matching of G if every element of M is an edge of G and any two edges of M are equal or share no vertex (no vertex is covered twice)
SimpleGraph.IsMaximalEdgeMatching A matching M of G is maximal if no further edge of G can be added to M while keeping the matching property
SimpleGraph.matching_card_bound Documented declaration: matching_card_bound
SimpleGraph.IsPathCover A family of paths covering all vertices without overlaps
SimpleGraph.pathCoverNumber Minimum size of a path cover of G
SimpleGraph.boxProd.decidableRel Adjacency in a box product of graphs with decidable adjacency is decidable
SimpleGraph.IsQuasiLineGraph A graph is a quasi-line graph if the neighborhood of every vertex can be partitioned into two cliques
SimpleGraph.HasFiniteRamseyProperty The finite Ramsey property for the pair
SimpleGraph.HasCountableRamseyEscape The countable Ramsey escape property for the pair
SimpleGraph.IsErdosHajnalExceptional A pair $(G_1, G_2)$ is Erdős–Hajnal exceptional if it has both the finite Ramsey property and the countable Ramsey escape property
SimpleGraph.graphRamsey Documented declaration: graphRamsey
SimpleGraph.diagonalGraphRamsey The diagonal graph Ramsey number R(G, G)
SimpleGraph.classicalRamsey The classical two-color Ramsey number R(k, l) = R(K_k, K_l)
SimpleGraph.diagonalRamsey The diagonal classical Ramsey number R(k) = R(K_k, K_k)
SimpleGraph.IsRamseySizeLinear Documented declaration: IsRamseySizeLinear
SimpleGraph.havelHakimiStep Helper function: Performs one step of the Havel-Hakimi reduction on a degree sequence
SimpleGraph.havelHakimiStep_length_cons havelHakimiStep drops the list length by exactly one on a nonempty list: (havelHakimiStep (d :: rest)).length = rest.length
SimpleGraph.residueAux Auxiliary function to calculate the residue recursively
SimpleGraph.residue Documented declaration: residue
SimpleGraph.saturationNumber The saturation number μ*(G) (also called the lower matching number): the minimum number of edges in a maximal matching of G
SimpleGraph.computableSaturationNumber Computable saturation number via powerset enumeration over the edge set
SimpleGraph.sizeRamsey The size Ramsey number r̂(G, H) is the minimum number of edges in a graph F such that any 2-coloring of F's edges contains a copy of G in one color or H in the other
SimpleGraph.Ls Ls G is the maximum number of leaves over all spanning trees of G
SimpleGraph.IsUniqueSubgraph G is a unique subgraph of H if there is exactly one subgraph of H isomorphic to G
SimpleGraph.uniqueSubgraphCount The number of distinct unique subgraphs of H: the number of isomorphism classes of graphs that occur exactly once as a subgraph of H
SimpleGraph.szeged_aux Auxiliary function for Szeged index: counts vertices closer to u than v
SimpleGraph.szegedIndex The Szeged index of G
SimpleGraph.computable_szeged_aux Computable Szeged auxiliary: count vertices closer to u than v
SimpleGraph.computable_szeged_index Computable Szeged index
SimpleGraph.temp_v temp_v G v = deg(v)/(n(G) - deg(v))
SimpleGraph.MaxTemp Maximum of temp_v over all vertices
SimpleGraph.UnitDistancePlaneGraph A unit distance graph in ℝ²: A graph where the vertices V are a collection of points in ℝ² and there is an edge between two points if and only if the distance between them is 1
SimpleGraph.IntegerDistancePlaneGraph Documented declaration: IntegerDistancePlaneGraph
SimpleGraph.UnitDistanceEmbeddable Documented declaration: UnitDistanceEmbeddable
SimpleGraph.HasDimension G has dimension n: the least m for which G embeds in ℝ^m with every edge a unit line segment
SimpleGraph.distToSet Distance from a vertex to a finite set
SimpleGraph.averageDistance Average distance of G
SimpleGraph.isInducedPath Check if a list of vertices forms an induced path in G
SimpleGraph.path The path number of a graph: The number of vertices of a largest induced path of the graph
SimpleGraph.length_le_path_of_isInducedPath An explicit induced path witnesses a lower bound for path
SimpleGraph.path_le_of_isInducedPath_length_le A uniform upper bound on the lengths of induced paths bounds path
SimpleGraph.path_le_of_not_exists_inducedPath_succ If G has no induced path on k + 1 vertices, then path G ≤ k
SimpleGraph.ecc Auxiliary quantity ecc used in conjecture 34
SimpleGraph.distMin The minimum distance between distinct vertices of $S$: $\min \{\operatorname{dist}_G(u, v) \mid u, v \in S, u \ne v\}$
SimpleGraph.eccSet The eccentricity of a set S: the maximum, over all vertices v of G, of the minimum distance from v to any vertex in S
SimpleGraph.distMaxSet The maximum distance between two vertices of a set S: $\operatorname{dist}_{\max}(S) = \max\{\operatorname{dist}_G(u,v) \mid u, v \in S\}$
SimpleGraph.distAvgSet The average distance between distinct vertices of a set S
SimpleGraph.distavg Average distance from all vertices to a given set
SimpleGraph.graphSquare The square of a graph G, denoted G²: two distinct vertices are adjacent iff their distance in G is at most 2
SimpleGraph.isInducedC4 Check whether four distinct vertices form an induced 4-cycle in G
SimpleGraph.countInducedC4 Count of induced C₄ subgraphs of G
SimpleGraph.computable_dist Computable graph distance via BFS
SimpleGraph.computableDistMin A computable version of distMin for a Finset of vertices
SimpleGraph.computable_avg_dist Computable average distance as a rational
SimpleGraph.distMin_eq_computableDistMin computableDistMin agrees with distMin on a finite vertex set
SimpleGraph.distEven distEven G v counts the vertices at even distance from v in G
SimpleGraph.evenDistancePairs The set of pairs of distinct vertices with even distance > 0
SimpleGraph.minEvenDistance Minimum even distance between distinct vertices in G
SimpleGraph.maxEvenDistance Maximum even distance between distinct vertices in G
SimpleGraph.averageEvenDistance Average even distance between distinct vertices in G
SimpleGraph.IsTotalDominatingSet A set S is a total dominating set of G if every vertex has a neighbor in S
SimpleGraph.IsMinimalTotalDominatingSet A total dominating set S is minimal if no proper subset of S is also a total dominating set
SimpleGraph.IsWellTotallyDominated A graph G is well totally dominated if every minimal total dominating set has the same cardinality
SimpleGraph.pendantVertices The set of pendant vertices (leaves) of G: vertices of degree 1
SimpleGraph.wienerIndex The Wiener index of G, which is the sum of distances between all pairs of vertices
SimpleGraph.computable_wiener Computable Wiener index: half the sum of all ordered pairwise distances
SimpleGraph.double_sum_eq_two_mul_sym2_sum Helper: ∑∑ f = 2 * ∑ Sym2 (lift f) when f symmetric with f(a,a) = 0
YoungDiagram.toSimpleGraph The simple graph of a Young diagram: two distinct cells are adjacent iff they lie in the same row or in the same column
AutomaticSequence.kKernel The k-kernel of a sequence a is the set of subsequences $(a_{k^e n + r})_{n \ge 0}$ with $e \ge 0$ and $0 \le r < k^e$
AutomaticSequence.self_mem_kKernel A sequence is a member of its own k-kernel, via e = r = 0
AutomaticSequence.IsAutomatic A sequence a is k-automatic if its k-kernel is finite
AutomaticSequence.IsAutomatic.finite_range A k-automatic sequence with k ≥ 2 takes finitely many values: a m is the value at 0 of the kernel element n ↦ a (k ^ m * n + m)
AutomaticSequence.kKernel_const The k-kernel of a constant sequence is a singleton
AutomaticSequence.isAutomatic_const A constant sequence is k-automatic
BitstringEncoding A canonical encoding of a type as bitstrings (List Bool)
BitstringEncoding.bitEncode The encoding function of the canonical BitstringEncoding of α
BitstringEncoding.bitDecode The decoding function of the canonical BitstringEncoding of α
BitstringEncoding.bitDecode_bitEncode Decoding is a left inverse of encoding
BitstringEncoding.ofLeftInverse Transport a BitstringEncoding along an injection f with partial inverse g
BitstringEncoding.delimit Make a bitstring self-delimiting: each payload bit b becomes the two bits [true, b], and the block is terminated by false
BitstringEncoding.undelimit Parse one self-delimiting block from the front of the input, returning the payload and the remaining input
BitstringEncoding.undelimitBlocksAux Parse a sequence of self-delimiting blocks, using fuel to bound the number of blocks
BitstringEncoding.undelimitBlocks Parse a sequence of self-delimiting blocks off the front of the input
BitstringEncoding.ofSubtype A subtype inherits the encoding of the ambient type; decoding additionally checks the defining predicate
ComplexityTheory.DecisionProblem The type of decision problems
ComplexityTheory.DecisionComplexityClass The type of complexity classes
ComplexityTheory.IsPolyTimeWithEncoding IsPolyTimeWithEncoding ea eb f asserts that f is computable in polynomial time when its input and output are encoded via the given Encodings ea and eb
ComplexityTheory.IsPolyTime Documented declaration: IsPolyTime
ComplexityTheory.isPolyTime_id The identity function is polynomial-time computable
ComplexityTheory.P The class P is the set of decision problems decidable in polynomial time by a deterministic Turing machine
ComplexityTheory.NP The class NP is the set of decision problems
ComplexityTheory.coNP The class coNP is the set of decision problems whose complements are in NP
ComplexityTheory.BPP The class BPP is the set of decision problems
DFA.IsSynchronizingWord A word w is a synchronizing word (or reset word) for a DFA M if reading w from any state leads to the same single state
DFA.IsSynchronizing A DFA M is synchronizing if it has at least one synchronizing word
DFA.isSynchronizingWord_nil The empty word is a synchronizing word for M exactly when all states of M collapse to a single one (equivalently, σ is a nonempty subsingleton)
DFA.IsSynchronizingWord.append Documented declaration: append
DFA.isSynchronizing_of_subsingleton A DFA whose state type is a nonempty subsingleton (i.e. a single-state DFA) is trivially synchronizing: the empty word already synchronizes it
Primcodable.ofLeftInverse If f : β → α has a left inverse finv such that f ∘ finv is primitive recursive, then β is Primcodable, with b encoded as Encodable.encode (f b)
Turing.BusyBeaver.Stmt A Turing machine "statement" is just a command to write a symbol on the tape (at the current position) and then move left or right
Turing.BusyBeaver.Machine Documented declaration: Machine
Turing.BusyBeaver.Cfg The configuration state of a Turing machine during operation consists of a label (machine state), and a tape
Turing.BusyBeaver.Machine.step Execution semantics of the Turing machine
Turing.BusyBeaver.Machine.Reaches The statement Reaches M s₁ s₂ means that s₂ is obtained starting from s₁ after a finite number of steps from s₂
Turing.BusyBeaver.Machine.init The initial configuration
Turing.BusyBeaver.Machine.eval Evaluate a Turing machine on initial input to a final state, if it terminates
Turing.BusyBeaver.Machine.IsHaltingInput M.IsHaltingInput l is the predicate that M is a halting configuration for M
Turing.BusyBeaver.Machine.IsHaltingConfiguration M.HaltsAtConfiguration s is the predicate that M is a halting configuration for M
Turing.BusyBeaver.Machine.IsHalting The property that a Turing Machine M eventually halts when starting from an empty tape
Turing.BusyBeaver.Machine.HaltsAfter The predicate that a machine starting at configuration s stops after at most n steps, i.e. reaches a configuration from which there are no defined transitions
Char.toDirSyntax Char.toDirSyntax c outputs the syntax corresponding to the character c if c is R or L and throws an error in other cases
Char.toNumeralSyntax Char.toBinarySyntax c outputs the syntax corresponding to the character c if c is in 0, ..., 9 and throws an error in other cases
Char.toStateSyntax Char.toStateSyntax c stateName outputs the syntax of the constructor corresponding to c if c is a capital letter (i.e. something between A and Z.)
Nat.toStateSyntax Nat.toStateSyntax n stateName outputs the syntax of the n-th constructor of the type used to index states
String.toStmtSyntax String.toStmtSyntax s stateName parses a component of a Turing machine string representation and outputs the syntax of the corresponding (state, statement) : State × Stmt pair
mkMachineMatchAltExpr Documented declaration: mkMachineMatchAltExpr
Nat.toStateName The name of the state type with n constructors A, B, ...
mkStateType Documented declaration: mkStateType
parseTuring parseTuring tmStringDescription outputs the Expr corresponding to the Turing Machine described by the string tmStringDescription
StateTransition.eval_get_eval TODO(Paul-Lez): also prove this for PFun.fix/golf using the PFun.fix API
StateTransition.eval_eq_eval TODO(Paul-Lez): also prove this for PFun.fix/golf using the PFun.fix API
StateTransition.eval_dom_iff TODO(lezeau): this should be generalized to PFun.fix
Finset.filter_subset_powersetCard_card_add_one The (#t + 1)-subsets of s that contain t ⊆ s are exactly the sets insert a t with a ∈ s \ t
Finset.card_filter_subset_powersetCard_card_add_one A k-subset t of s lies in exactly #s - k of the (k + 1)-subsets of s
Finset.reciprocalSum The reciprocal sum of a finite set of natural numbers
Int.exists_eq_of_natAbs_sub_le_one Discrete intermediate value theorem
Nat.Full A natural number $n$ is said to be $k$-full (or $k$-powerful) if for every prime factor $p$ of $n$, the $k$-th power $p^k$ also divides $n$
Nat.Full.zero_left Every natural number is $0$-full
Nat.Full.one_left Every natural number is $1$-full
Nat.Full.zero_right $0$ is always $k$-full
Nat.Full.one_right $1$ is always $k$-full
Nat.Powerful A Powerful number is a natural number $n$ where for every prime divisor $p$, $p^2$ divides $n$
Nat.not_full_of_prime_mod_prime_sq If $n \equiv p \pmod{p ^ (k + 1)}$, for a prime $p$ then $n$ is not $(k + 1)$-full
Nat.maxPrimeFac The greatest prime divisor of a natural number n > 1
Nat.maxPrimeFac_dvd The greatest prime factor of a natural number divides it
Nat.le_maxPrimeFac Every prime factor of a nonzero natural number is at most its greatest prime factor
Nat.Prime.maxPrimeFac_eq_self The greatest prime factor of a prime is the prime itself
Nat.maxPrimeFac_eq_self_iff The fixed points of maxPrimeFac are zero, one, and the primes
Nat.maxPrimeFac_mul The greatest prime factor of a product of nonzero natural numbers is the maximum of their greatest prime factors
Nat.maxPrimeFac_pow The greatest prime factor of a nonzero power is the greatest prime factor of its base
Nat.maxPrimeFac_le The greatest prime factor of a natural number is at most that number
Nat.isLeast_maxPrimeFac The greatest prime factor of a natural number greater than one is the least upper bound of its prime factors
Nat.maxPrimeFac_eq_sSup Away from n = 1, the computable greatest prime factor agrees with its supremum characterization
Nat.IsPerfectPower Documented declaration: IsPerfectPower
Finset.Coprime A Finset of numbers is coprime, or relatively prime, if its gcd is 1
Finset.Coprime_pair_iff A Finset of numbers is coprime, or relatively prime, if its gcd is 1
Nat.squarefreePart n.squarefreePart is the unique a₀ : ℕ such that a₀ is squarefree and n = a₀ * b ^ 2, for some b : ℕ
Nat.squarefreePart_of_squarefree If n is squarefree, then its squarefree part is itself
Nat.squarefreePart_of_isSquare The squarefree part of any square is 1
Nat.squarefreePart_factorization If $n = \prod_p p^{e_p}$ is the prime factorization of $n$, then $\prod_p p^{e_p \pmod{2}}$ is the prime factorization of the squarefree part of $n$
Nat.squarePart The square part is the value of b ^ 2 in the squarefree decomposition of n = a₀ * b ^ 2
Nat.squarefreePart_mul_squarePart The squarefree decomposition of a natural number
gompertzConstant Gompertz constant $$\delta = -e * \int_1^∞ e^{-t}/t dt \approx 0.59634$$
catalanConstant Catalan's constant $$G = \sum_{n=0}^∞ (-1)^n / (2n + 1)^2 \approx 0.91596$$
distToNearestInt The distance from a real number to the nearest integer
distToNearestInt_nonneg The distance to the nearest integer is nonnegative
Set.partialDensity Documented declaration: partialDensity
Set.upperDensity Documented declaration: upperDensity
Set.lowerDensity Documented declaration: lowerDensity
Set.HasDensity Documented declaration: HasDensity
Set.HasPosDensity Documented declaration: HasPosDensity
Set.intPartialDensity The two-sided partial natural density of a set of integers, counted inside the interval [-N, N]
Set.HasIntDensity A set of integers has two-sided natural density α
Set.HasDensity.univ In a non-trivial partial order with a least element, the set of all elements has density one
Nat.hasDensity_even The natural density of the set of even numbers is 1 / 2
Nat.hasDensity_zero_of_finite A finite set has natural density zero
Nat.infinite_of_hasDensity_pos A set of positive natural density is infinite
Set.HasLogDensity A set A of natural numbers has logarithmic density d if the sequence $(1 / \log n) \cdot \sum_{k \in A, k \le n} (1/k)$ converges to d
Set.Triplewise The ternary relation r holds triplewise on the set s if r x y z for all distinct x y z ∈ s
𝔽 𝔽 p n is the vector space (Z/pZ)^n
𝔽₂ 𝔽₂ n is the vector space $\mathbb{F}_2^n$
𝔽₃ 𝔽₃ n is the vector space $\mathbb{F}_3^n$
𝔽₅ 𝔽₅ n is the vector space $\mathbb{F}_5^n$
𝔽₇ 𝔽₇ n is the vector space $\mathbb{F}_7^n$
IsPerfectDifferenceSet A perfect difference set modulo n is a set D such that the map (a, b) ↦ a - b from D.offDiag to {x : ZMod n | x ≠ 0} is a bijection
SymbolicDynamics.blockComplexity The block complexity $p(n, a)$ of a sequence $a$, that is, the number of distinct blocks $a_k a_{k+1} \cdots a_{k+n-1}$ of $n$ consecutive terms of $a$
SymbolicDynamics.blockEntropy The entropy of a sequence
SymbolicDynamics.blockComplexity_const A constant sequence has exactly one block of each length
SymbolicDynamics.blockEntropy_const A constant sequence has zero entropy
Field.absoluteGaloisGroup.mapOfAlgebra A commuting square of two fields and their algebraic closures induces a continuous homomorphism of their absolute Galois groups
Field.absoluteGaloisGroup.map An embedding of fields induces a continuous homomorphism of absolute Galois groups
MilnorK.Rel The relations defining K^M_*(F)/ℓ as a quotient of the tensor algebra of Fˣ: the Steinberg relation {a}{b} = 0 when a + b = 1, and the relation ℓ = 0
MilnorK The Milnor K-ring K^M_*(F)/ℓ of the field F with coefficients in ℤ/ℓ
MilnorK.of The class {a} of a unit a in K^M_1(F)/ℓ
MilnorK.ofHom The class map {·} : Fˣ → K^M_1(F)/ℓ as an additive homomorphism
MilnorK.of_mul_of_eq_zero The Steinberg relation {a}{b} = 0 when a + b = 1
MilnorK.natCast_eq_zero The relation ℓ = 0 in K^M_*(F)/ℓ
MilnorK.symbol The symbol {a 0, …, a (n - 1)} = {a 0} ⋯ {a (n - 1)} of an n-tuple of units
MilnorK.symbol_pair_eq_zero The Steinberg relation {a, b} = 0 when a + b = 1
MilnorK.grade The degree-n part K^M_n(F)/ℓ of K^M_*(F)/ℓ: the additive subgroup generated by the symbols of degree n
MilnorK.symbolLengthBounds The set of m such that every element of K^M_n(F)/ℓ is a sum of at most m symbols
MilnorK.grade_eq_bot_of_forall_exists_pow_eq If every unit of F is an ℓ-th power, then K^M_n(F)/ℓ = 0 for n ≥ 1
MilnorK.augmentation The augmentation K^M_*(F)/ℓ → ℤ/ℓ, which sends every class {a} to 0
MilnorK.nontrivial K^M_*(F)/ℓ is nontrivial for ℓ ≠ 1, since the augmentation maps it onto ℤ/ℓ
MilnorK.one_mem_grade_zero The empty symbol is 1, so 1 ∈ K^M_0(F)/ℓ
MilnorK.mem_grade_zero_iff K^M_0(F)/ℓ = ℤ/ℓ is generated by the empty symbol 1
MilnorK.sub_one_le_of_mem_symbolLengthBounds_zero Documented declaration: sub_one_le_of_mem_symbolLengthBounds_zero
MilnorK.sub_one_mem_symbolLengthBounds_zero Every element of K^M_0(F)/ℓ = ℤ/ℓ is a sum of at most ℓ - 1 empty symbols
MilnorK.isLeast_symbolLengthBounds_zero For ℓ ≥ 1, the symbol length of K^M_0(F)/ℓ = ℤ/ℓ is ℓ - 1
MilnorK.toDualNumber Documented declaration: toDualNumber
MilnorK.of_ne_zero Degree one does not collapse: if a homomorphism φ : Fˣ → ℤ/ℓ is nonzero on a, then the class {a} is nonzero in K^M_1(F)/ℓ
MilnorK.grade_one_ne_bot K^M_1(F)/ℓ ≠ 0 whenever some homomorphism φ : Fˣ → ℤ/ℓ is nonzero
MilnorK.grade_one K^M_1(F)/ℓ is the image of Fˣ, since sums and negatives of degree-one symbols are again degree-one symbols
MilnorK.one_mem_symbolLengthBounds_one Every class in K^M_1(F)/ℓ is a single symbol
MilnorK.zero_notMem_symbolLengthBounds_one 0 is not a bound for the symbol length in degree 1 when K^M_1(F)/ℓ ≠ 0
MilnorK.isLeast_symbolLengthBounds_one The symbol length of K^M_1(F)/ℓ is 1 as soon as the group is nonzero
MvRatFunc Documented declaration: MvRatFunc
IsDefined A rational function $g$ is defined at a point $x_0$ if $g$ can be written as $p / q$ where $p, q$ are polynomials and $q(x_0) \neq 0$
Module.orientedEuclideanSpaceFinTwo Oriented angles make sense in 2d
fact_finrank_euclideanSpace_fin_two Two dimensional euclidean space is two-dimensional
EuclideanGeometry.NonTrilinear We say a subset A of points in the plane is non-trilinear if it contains no three points that lie on the same line
EuclideanGeometry.NonCollinearFor We say a subset S of points is non-collinear for $n$ points if it contains no $n$ points that lie on the same line
EuclideanGeometry.ConvexIndep ConvexIndep S means that S consists of extremal points of its convex hull, i.e., the point set encloses a convex shape
EuclideanGeometry.HasConvexNGon The set P contains a convex n-gon
EuclideanGeometry.IsCcwConvexPolygon The statement that a sequence of points form a counter-clockwise convex polygon
EuclideanGeometry.IsConvexPolygon The statement that a sequence of points form a convex polygon
EuclideanGeometry.isConvexPolygon_three_of_affineIndependent Three affine independent points always form a convex polygon
EuclideanGeometry.maximalDistinctDistancesFrom Documented declaration: maximalDistinctDistancesFrom
EuclideanGeometry.InGeneralPosition A collection $x_1, \dots, x_n\in\mathbb{R}^2$ is in general position if no three are collinear and no four lie on a circle
EuclideanGeometry.IsRightAngled a b c are the vertices of a right-angled triangle: the (unoriented) angle at one of the three vertices equals π / 2
EuclideanGeometry.IsIsoscelesTrapezoid Documented declaration: IsIsoscelesTrapezoid
EuclideanGeometry.IsLine A line in the plane: an affine subspace of ℝ² whose direction is one-dimensional
Set.IsIsoscelesFree A set is isosceles-free if no three distinct points in it form an isosceles triangle
Module.orientedEuclideanSpaceFinThree The standard basis gives us a preferred orientation in ℝ³
fact_finrank_euclideanSpace_fin_three Three dimensional euclidean space is three-dimensional
Group.Generators.IsHyperbolicWith A group is δ-hyperbolic with respect to a generating family P if its induced metric space is δ-hyperbolic
Group.IsHyperbolic A group G is hyperbolic if there exists a finite generating family P and a constant δ such that G is δ-hyperbolic with respect to P
Group.Generators.wordLength The word length of g with respect to the generating family P
Group.Generators.IsGeodesic A word l is geodesic if its length is exactly the word length of the group element it represents
Group.Generators.exists_isGeodesic Every group element has a geodesic word representative
Group.Generators.wordLength_le_iff The characterisation of word length: the word length of a group element is less or equal to n if and only if there exists a word l of length at most n which evaluates to g
Group.Generators.wordLength_inv_le This is superseded by wordLength_inv
Group.Generators.groupNorm The wordLength with respect to a generating family P defines a norm on G
Group.Generators.normedGroup G as a metric space NormedGroup G with respect to a generating family P
Group.Generators.wordProd The canonical map from a word List (ι × Bool) over a generating family ι to its corresponding group G
Group.Generators.wordProd_surjective Every element of G is the product of some word over a generating family
LieGroupPresentation A continuous group isomorphism from G to an n-dimensional real-analytic Lie group
AdmitsLieGroupStructure A topological group admits a Lie group structure if it has a LieGroupPresentation in some finite dimension
admitsLieGroupStructure_of_lieGroup Every Hausdorff finite-dimensional real-analytic Lie group admits a Lie group structure
admitsLieGroupStructure_of_discreteTopology Every discrete group is a 0-dimensional Lie group
AdmitsLieGroupStructure.locallyCompactSpace A group admitting a Lie group structure is locally compact
unitDistNum The number of pairs of points of a finite set s in a metric space that are distance 1 apart
distanceSet The set of distances determined by a finite set of points in a metric space
distinctDistances Given a finite set of points in a metric space, we define the number of distinct distances between pairs of points
minimalDistinctDistances The minimum number of distinct distances determined by a set of n points in X
distanceMultiplicity The multiplicity of the distance d determined by points, that is, the number of unordered pairs of distinct points at distance d apart
distinctDistancesFrom Given a finite set of points in a metric space, we define the number of distinct distances between a given point and all other points
unitDistNum_le_choose_two The number of unit-distance pairs of a finite set of n points is at most $\binom{n}{2}$, the total number of unordered pairs of distinct points
even_card_offDiag_filter_dist_eq The ordered pairs of distinct points of points at distance d come in swapped pairs, so there are evenly many of them
sum_distanceMultiplicity The multiplicities of the distances determined by points add up to the number of unordered pairs of distinct points
Monoid.IsTorsionFree A predicate on an additive monoid saying that only 0 has finite order
Monoid.IsTorsionFree.of_isMulTorsionFree Uniqueness of roots implies torsion-freeness
Lean.Elab.InfoTree.visitStateOf Visit nodes in an infotree, where state is set only within a single branch
Matrix.GeneralLinearGroup.diagonalHom The multiplicative group homomorphism (Rˣ)ⁿ →* GL(n, R) given by mapping a vector x to Matrix.diagonal x
Matrix.GeneralLinearGroup.diagonalSubgroup The group of invertible diagonal matrices
QuadraticMap.Anisotropic.radical_eq_bot The radical of an anisotropic quadratic form is trivial
QuadraticMap.Anisotropic.nondegenerate_of_rank_le_one An anisotropic quadratic form on a module of rank at most 1 is nondegenerate
QuadraticMap.Anisotropic.nondegenerate In characteristic not 2, an anisotropic quadratic form is nondegenerate
QuadraticForm.anisotropicDims The set of n such that there is a nondegenerate anisotropic quadratic form on Fin n → F
QuadraticForm.mem_anisotropicDims_iff In characteristic not 2, the nondegeneracy condition in anisotropicDims is automatic
QuadraticForm.anisotropicDims_eq_univ Over a linearly ordered field, the sum of n squares is anisotropic for every n, so every n is the dimension of an anisotropic quadratic form
Matrix.SpecialLinearGroup.diagonalSubgroup The group of invertible diagonal matrices with determinant 1
FirstOrder.Language.Theory.IsRecursive Documented declaration: IsRecursive
FirstOrder.Language.Theory.IsDecidable Documented declaration: IsDecidable
FirstOrder.Language.Theory.isRecursive_empty The empty set of sentences is recursive
FirstOrder.Language.Theory.isRecursive_univ The set of all sentences is recursive
FirstOrder.Language.Theory.IsRecursive.compl The complement of a recursive set of sentences is recursive
FirstOrder.Language.Theory.isDecidable_congr Decidability only depends on the consequences of a theory
FirstOrder.Language.Theory.isDecidable_iff_isRecursive For a theory closed under consequence, decidability is the same as recursiveness
FirstOrder.Language.Theory.isDecidable_completeTheory_iff The complete theory of a structure M contains all its consequences, so it is decidable if and only if membership in it is computable
FirstOrder.Language.Theory.isDecidable_univ The inconsistent theory of all sentences is decidable
FirstOrder.Language.sum.encodableFunctions The function symbols of a sum of two languages are encodable when those of both summands are
FirstOrder.Language.sum.encodableRelations The relation symbols of a sum of two languages are encodable when those of both summands are
FirstOrder.Language.IsRelational.encodableFunctions A relational language has no function symbols
FirstOrder.Language.IsAlgebraic.encodableRelations An algebraic language has no relation symbols
FirstOrder.Language.ring.encodableFunctions The function symbols +, *, -, 0, 1 of the language of rings are encoded as 0, 1, 2, 3, 4
FirstOrder.Language.order.encodableRelations The unique relation symbol ≤ of the language of orders is encoded as 0
FirstOrder.Language.BoundedFormula.encodableSigma The Gödel numbering of bounded formulas, obtained from the list encoding BoundedFormula.encoding and the Encodable instances on lists and on the symbols
FirstOrder.Language.BoundedFormula.encodable The Gödel numbering of bounded formulas with n bound variables, and in particular of formulas (n = 0) and sentences (n = 0 and α = Empty)
FirstOrder.expFunc The type of function symbols of Language.exp: a single unary symbol exp
FirstOrder.Language.exp The language with a single unary function symbol exp and no relation symbols
FirstOrder.Language.orderedExpField The language of ordered exponential fields: the language of rings, together with a unary function symbol exp and the order relation ≤
FirstOrder.Language.orderedExpField.instIsOrdered The language of ordered exponential fields contains the order symbol ≤
FirstOrder.Language.exp.encodableFunctions The function symbol exp is encoded as 0
FirstOrder.Language.compatibleRingReal ℝ as a structure for the language of rings, with the usual ring operations
FirstOrder.Language.orderStructureReal ℝ as a structure for the language of orders, with ≤ interpreted as ≤
FirstOrder.Language.expStructureReal ℝ as a structure for Language.exp, with exp interpreted as the exponential function
FirstOrder.Language.orderedExpFieldOrderedStructureReal In the real exponential field, ≤ is interpreted as ≤
IsAdditionChain An addition chain is a strictly increasing sequence
additionChainLength The length $\ell(n)$ of $n$: the minimal number of addition steps (the number of entries minus one) over all addition chains ending at $n$
additionChainSteps The set of step counts realised by chains ending at n
additionChainLength_le Exhibiting a chain bounds ℓ above
IsAdditionChain.dropLast Dropping the last entry of a chain leaves a chain
IsAdditionChain.getLast_le_two_pow Every step at most doubles, so a chain of r steps cannot reach past 2 ^ r
le_two_pow_additionChainLength The doubling bound, transported to ℓ: reaching n takes at least log₂ n steps
lt_additionChainLength_of_two_pow_lt The lower-bound tool: r steps cannot reach past 2 ^ r
IsAdditiveComplement A and B are additive complements if A + B contains all large integers
subsetSums The set of subset sums of a set A ⊆ M
subsetSums_mono If A ⊆ B, then subsetSums A ⊆ subsetSums B
subseqSums' The set of subset sums of a sequence ℕ → M, where repetition is allowed
IsAddComplete A set A ⊆ M is complete if every sufficiently large element of M is a subset sum of A
IsAddComplete.mono If A ⊆ B and A is complete, then B is also complete
IsAddStronglyComplete A set A ⊆ M is complete if every sufficiently large element of M is a subset sum of A
IsAddStronglyComplete.isAddComplete A strongly complete set is complete
IsAddStronglyComplete.mono If A ⊆ B and A is strongly complete, then B is also strongly complete
IsAddStronglyCompleteNatSeq A sequence A is strongly complete if fun m => A (n + m) is still complete for all n
IsAddStronglyCompleteNatSeq.isAddComplete A strongly complete sequence is complete
IsAddStronglyCompleteNatSeq.of_isAddStronglyComplete If the range of a sequence A is strongly complete, then A is strongly complete
IsAddStronglyCompleteNatSeq.isAddStronglyComplete If A is strongly complete and the preimage of each element is finite, then the range of A is strongly complete
IsAddCompleteNatSeq' A sequence A is complete if every sufficiently large element of M is a sum of (not necessarily distinct) terms of A
not_isAddComplete_range_diff_of_sum_range_le Let A : ℕ → ℕ be strictly increasing from index n₀ on, with A 0 + ⋯ + A (m - 1) ≤ A (m + 1) for all m ≥ n₀
IsAmicable We say that $a,b\in \mathbb{N}$ are an amicable pair if $\sigma(a)=\sigma(b)=a+b$
AutomaticSequence.IsAutomaticInBase The base-b expansion of ξ is generated by a finite automaton, in the classical sense: its digit sequence is b-automatic
AutomaticSequence.IsBroadlyAutomaticInBase The base-b expansion of ξ is generated by a finite automaton, in the broad sense: its digit sequence is k-automatic for some k ≥ 2, not necessarily for k = b
AutomaticSequence.IsAutomaticInBase.broad The classical reading implies the broad one, by taking k = b
AutomorphicForm.rightTranslateSpan The k-span of the right translates of f by the subgroup K
AutomorphicForm.IsKFinite f is K-finite: the span of its right K-translates is finite-dimensional
AutomorphicForm.isKFinite_of_rightInvariant A function invariant under right translation by K is K-finite
AutomorphicForm.isKFinite_of_finite If K is finite then every function is K-finite
AutomorphicForm.IsKFinite.add K-finiteness is closed under addition: the translate span of f + g sits inside the sum of the translate spans
AutomorphicForm.IsZFinite m is Z-finite: it is annihilated by an ideal of Z of finite codimension over k
AutomorphicForm.IsZFinite.add Z-finiteness is closed under addition: the intersection of the two annihilating ideals works, since Z ⧸ (I ⊓ J) embeds in (Z ⧸ I) × (Z ⧸ J)
AutomorphicForm.IsZFinite.const_smul Z-finiteness is preserved by scalar multiplication: the same ideal works
AutomorphicForm.IsZFinite.of_forall_smul_eq_algHom_smul An element on which Z acts through a k-algebra character is Z-finite: the kernel of the character is an ideal of finite codimension annihilating it
AutomorphicForm.IsZFinite.pi_const A constant family with Z-finite value is Z-finite, with the same ideal
AutomorphicForm.IsZFinite.comp Precomposition preserves Z-finiteness of a family, with the same ideal
Matrix.GeneralLinearGroup.entrySup The sup norm on the entries of a matrix: for n = 2 this is |(a b; c d)| = max {|a|, |b|, |c|, |d|}
Matrix.GeneralLinearGroup.gnorm The norm ‖y‖ = max (|y|, |y⁻¹|) on GL n ℝ, with |·| the sup norm on matrix entries
Matrix.GeneralLinearGroup.inv_card_le_gnorm gnorm is uniformly bounded below: since y * y⁻¹ = 1, the entries of y and y⁻¹ cannot all be small
Matrix.GeneralLinearGroup.IsSlowlyIncreasing A function φ : GL n ℝ → ℂ is slowly increasing, or of moderate growth, if ‖φ y‖ ≤ C * ‖y‖ ^ r for some real C and natural number r
Matrix.GeneralLinearGroup.exists_forall_norm_le_pow_of_le Documented declaration: exists_forall_norm_le_pow_of_le
Matrix.GeneralLinearGroup.isZFinite_oneSmoothGL The constant function 1 is Z(𝔤)-finite: the kernel of constantsCharacter is an ideal of finite codimension annihilating it
Matrix.GeneralLinearGroup.isZFinite_const_smoothGL Every constant function is Z(𝔤)-finite
Matrix.GeneralLinearGroup.IsSmoothAdelic Smoothness of a function on G(𝔸) = G(𝔸_f) × G(ℝ), with G(𝔸_f) = GL n 𝔸ᶠ[ℤ, ℚ]: continuous, locally constant in the finite variable, and C^∞ in the archimedean variable
Matrix.GeneralLinearGroup.IsAutomorphicForm Documented declaration: IsAutomorphicForm
Matrix.GeneralLinearGroup.isAutomorphicForm_const Sanity check: the constant functions are automorphic forms
Matrix.GeneralLinearGroup.isAutomorphicForm_one Sanity check: the constant function 1 is an automorphic form
Matrix.GeneralLinearGroup.automorphicForms The automorphic forms for GL n / ℚ as a ℂ-submodule of the functions on G(𝔸)
Matrix.GeneralLinearGroup.IsAutomorphicForm.rightTranslate Automorphy is preserved by right translation in the finite variable
Matrix.GeneralLinearGroup.rightTranslation The right translation representation of G(𝔸_f) on the automorphic forms: g acts by f ↦ fun (x, y) => f (x * g, y)
Matrix.GeneralLinearGroup.universalEnveloping The universal enveloping algebra U(𝔤𝔩 n ℂ) of the complexified Lie algebra of GL n ℝ
Matrix.GeneralLinearGroup.centerUniversalEnveloping The centre of the universal enveloping algebra of the complexified Lie algebra of GL n ℝ
Matrix.GeneralLinearGroup.lieDerivHom The action of 𝔤𝔩 n ℂ on the C^∞ functions on GL n ℝ by left invariant differential operators, as a homomorphism of ℂ-Lie algebras
Matrix.GeneralLinearGroup.envelopingAction The action of the universal enveloping algebra U(𝔤𝔩 n ℂ) on the C^∞ functions on GL n ℝ, obtained from lieDerivHom by the universal property
Matrix.GeneralLinearGroup.centerAction The centre of the universal enveloping algebra acting on the C^∞ functions on GL n ℝ
Matrix.GeneralLinearGroup.instModuleCenterSmoothGL The C^∞ functions on GL n ℝ as a module over the centre of the universal enveloping algebra, via left invariant differential operators
Matrix.GeneralLinearGroup.oneSmoothGL The constant function 1 as an element of smoothGL n; the constant functions are the line it spans
Matrix.GeneralLinearGroup.eq_smul_oneSmoothGL_of_mem_span An element of the line of constant functions is determined by its value at 1
Matrix.GeneralLinearGroup.envelopingAction_mem_span_oneSmoothGL Left invariant differential operators map the constant functions to constant functions: the generators lieDerivC X kill them
Matrix.GeneralLinearGroup.constantsCharacter The character by which the centre of the enveloping algebra acts on the constant functions: z • 1 = constantsCharacter n z • 1
AutomorphicForm.LieDerivAux.mulRightL Right multiplication by X, as a continuous linear map
AutomorphicForm.LieDerivAux.fderiv_rightDeriv_apply Documented declaration: fderiv_rightDeriv_apply
AutomorphicForm.LieDerivAux.fderiv_rightDeriv_sub_comm The commutator of two left invariant derivatives is the left invariant derivative along the commutator: the second-order terms cancel by symmetry of the second derivative
Matrix.GeneralLinearGroup.IsSmoothOnGL φ : GL n ℝ → ℂ is C^∞: it extends to a C^∞ function on the open set of invertible matrices
Matrix.GeneralLinearGroup.isOpen_setOf_isUnit The invertible matrices are an open set: they are the nonvanishing locus of det
Matrix.GeneralLinearGroup.extendGL A choice of C^∞ extension of φ to the invertible matrices, when one exists
Matrix.GeneralLinearGroup.fderiv_extendGL_eq The derivative of the chosen extension at an invertible matrix does not depend on the choice: any two extensions of φ agree on the open set of invertible matrices
Matrix.GeneralLinearGroup.expGL The exponential of a matrix, as an element of GL n ℝ: exp X is invertible with inverse exp (-X)
Matrix.GeneralLinearGroup.lieDerivFun Documented declaration: lieDerivFun
Matrix.GeneralLinearGroup.hasDerivAt_lieDerivFun Differentiating along the one-parameter subgroup computes the derivative of any smooth extension in the direction y * X
Matrix.GeneralLinearGroup.lieDerivFun_eq_fderiv The one-parameter-subgroup description of the left invariant derivative agrees with the directional-derivative one
Matrix.GeneralLinearGroup.smoothGL The ℂ-submodule of C^∞ functions on GL n ℝ, which is what the Lie algebra and hence the universal enveloping algebra acts on
Matrix.GeneralLinearGroup.lieDeriv The left invariant derivative along X as a ℂ-linear endomorphism of the C^∞ functions on GL n ℝ
Matrix.GeneralLinearGroup.lieDerivFun_bracket The commutator of two left invariant derivatives is the left invariant derivative along the commutator of the directions
Matrix.GeneralLinearGroup.map_re_bracket The real part of a complex commutator, arranged as a difference of two real commutators
Matrix.GeneralLinearGroup.map_im_bracket The imaginary part of a complex commutator, arranged as a sum of two real commutators
Matrix.GeneralLinearGroup.lieDerivC The action of the complexified Lie algebra 𝔤𝔩 n ℂ of GL n ℝ on the C^∞ functions: X + i Y acts as lieDeriv X + i • lieDeriv Y
Matrix.GeneralLinearGroup.diagonalEmbedding The diagonal embedding of the rational points G(ℚ) = GL n ℚ into G(𝔸_f) × G(ℝ) = GL n 𝔸ᶠ[ℤ, ℚ] × GL n ℝ, through the inclusions of ℚ into the finite adeles and into ℝ
Matrix.GeneralLinearGroup.ratDiagonal The subgroup Γ = G(ℚ) of G(𝔸_f) × G(ℝ): the range of the diagonal embedding of GL n ℚ
Matrix.GeneralLinearGroup.orthogonalSubgroup Documented declaration: orthogonalSubgroup
Matrix.GeneralLinearGroup.abs_coe_le_one_of_mem_orthogonalSubgroup The entries of an orthogonal matrix are bounded by 1: each row is a unit vector
Matrix.GeneralLinearGroup.integralAdeles The integral adeles Ẑ = ∏ᵥ ℤᵥ as a subring of the finite adeles of ℚ: the adeles that are integral at every place
Matrix.GeneralLinearGroup.integralAdelicSubgroup GL n Ẑ inside GL n 𝔸ᶠ[ℤ, ℚ]: the matrices whose entries, and whose inverse's entries, are integral adeles
IsBeurlingPrimes A sequence of real numbers 1 < a 0 < a 1 < ... is called a set of Beurling prime numbers if it tends to infinity
beurlingInteger A Beurling integer is a number of the form ∏ i, (a i) ^ (k i) for a given sequence a and a finitely-supported sequence of naturals k
BeurlingIntegers The set of Beurling integers are numbers of the form ∏ i, (a i) ^ (k i), where k has finite support
generator_mem_beurling Every element of the sequence a is a Beurling integer
mul_mem_beurling The set of Beurling integers is closed under multiplication
pow_mem_beurling The set of Beurling integers is closed under taking powers
IsCarmichael A Carmichael number is a composite number n such that for all b ≥ 1, we have b^n ≡ b (mod n)
CoveringSystem A covering system of a semiring R is a finite set of cosets of non-zero ideals whose union covers the whole ring
StrictCoveringSystem We say a covering system is strict if all the congruence relations that define it are take modulo a different ideal (or number)
IsZNumber A Z-number is a positive real number ξ such that the fractional part of ξ (3/2)^n lies in [0, 1/2) for every positive integer n
Nat.nth_divisors_zero The smallest divisor of a positive number is 1, i.e. the 0th entry of the increasing enumeration of its divisors
Nat.two_le_nth_divisors Every divisor enumerated after index 0 is at least 2
Real.partQuot The sequence $(c_n)_{n \ge 1}$ of partial quotients of the continued fraction expansion $\xi = [c_0; c_1, c_2, \ldots]$, indexed from $0$
Real.cfEntropy The entropy $E(\xi)$ of the continued fraction expansion of $\xi$
Real.baseEntropy The entropy $E(\xi, b)$ of the base $b$ expansion of $\xi$
harmonicBlock The block harmonic sum $\sum_{n\leq k\leq m}\frac{1}{k}$
harmonicBlock_one The block sum starting at 1 is Mathlib's harmonic number
IsLacunary Say a sequence is lacunary if there exists some $\lambda > 1$ such that $a_{n+1}/a_n > \lambda$ for all sufficiently large $n$
IsLacunary.eventually_lt Every lacunary sequence is eventually strictly increasing
IsLacunaryReal The real-valued analogue of IsLacunary
isLacunary_iff_isLacunaryReal An ℕ-valued sequence is lacunary iff its real cast is lacunary as a real sequence
IsLacunaryReal.eventually_lt A positive lacunary real-valued sequence is eventually strictly increasing
Nat.MultiplicativelyIndependent Two integers $p, q \ge 2$ are multiplicatively independent if $\log p / \log q$ is irrational, equivalently if $p^m = q^n$ holds only for $m = n = 0$
Nat.MultiplicativelyIndependent.symm Multiplicative independence is symmetric
Nat.MultiplicativelyIndependentFamily A family of integers is multiplicatively independent if there is no nonzero tuple of integers
Nat.MultiplicativelyIndependentFamily.pairwise A multiplicatively independent family is pairwise multiplicatively independent
Nat.MultiplicativelyIndependent.family A multiplicatively independent pair is a multiplicatively independent family
NormalNumber.digitSeq The n-th digit (0-indexed) after the radix point in the base-b expansion of x
NormalNumber.IsSimplyNormalInBase A real number x is simply normal in base b if every digit d < b appears with asymptotic frequency 1 / b in the base-b fractional expansion of x
NormalNumber.IsNormalInBase Documented declaration: IsNormalInBase
NormalNumber.IsNormalInBase.isSimplyNormalInBase A number that is normal in base b is simply normal in base b
NormalNumber.IsAbsolutelyNormal A real number x is absolutely normal if it is normal in every integer base b ≥ 2
NumberField.IsFundamentalDiscr Fundamental discriminants are those integers D that appear as discriminants of quadratic fields
NumberField.isFundamentalDiscr_iff Documented declaration: isFundamentalDiscr_iff
QuadraticAlgebra.instIsQuadraticExtension Any QuadraticAlgebra ℚ a b that is a field is automatically a quadratic extension of ℚ, i.e., a degree-2 extension
Algebra.IsQuadraticExtension.to_numberField A quadratic field is a number field: it has characteristic zero and is finite-dimensional over ℚ
IsPisot Documented declaration: IsPisot
isPisot_goldenRatio Documented declaration: isPisot_goldenRatio
Nat.IsPractical A natural number $n$ is practical if every positive integer $m ≤ n$ can be represented as a sum of distinct divisors of $n$
primeGap The prime gap: the difference between the $n+1$-th and $n$-th prime
Set.IsPrimitive An element n is primitive in a set S ⊆ ℕ if n ∈ S and no proper divisor of n is in S
Set.primitives The set of primitive elements of S
Nat.IsSierpinskiNumber A positive odd integer $k$ is a Sierpiński number if $k \cdot 2^n + 1$ is composite for all natural numbers $n$
scaleL The scale $L(n)=\exp(\sqrt{\log n\log\log n})$
LucasSequence.U The Lucas sequence of the first kind $U_0(P, Q) = 0$, $U_1(P, Q)=1$, $U_{n+2}(P, Q)=PU_{n+1}(P, Q)-QU_n(P, Q)$
LucasSequence.V The Lucas sequence of the second kind $V_0(P, Q) = 0$, $V_1(P, Q)=P$, $V_{n+2}(P, Q)=PV_{n+1}(P, Q)-QV_n(P, Q)$
lucasNumber The Lucas numbers $L_0 = 2$, $L_1=1$, $L_{n+2} = L_{n+1}+L_n$
IsWallSunSunPrime Wall–Sun–Sun prime A prime $p$ is a Wall–Sun–Sun prime if and only if $L_p \equiv 1 \pmod{p^2}$, where $L_p$ is the $p$-th Lucas number
IsLucasWieferichPrime Documented declaration: IsLucasWieferichPrime
LucasSequence.dvd_U_two_mul The parameter $P$ divides every even-indexed term of the Lucas sequence $U(P, Q)$
IsLucasWieferichPrime.of_sq_dvd Documented declaration: of_sq_dvd
IsWieferichPrimeBase Wieferich prime to base a A prime $p$ is a Wieferich prime to base $a$ if $p^2 \mid a^{p-1} - 1$, i.e. if $a^{p-1} \equiv 1 \pmod{p^2}$
IsWieferichPrimeBase.of_modEq_one Every prime p is a Wieferich prime to any base a ≡ 1 [MOD p ^ 2]
isWieferichPrimeBase_iff_pow_modEq For a nonzero base a, being a Wieferich prime to base a is the congruence a ^ (p - 1) ≡ 1 [MOD p ^ 2]
isWieferichPrimeBase_zero_iff Every prime is a Wieferich prime to base 0, because 0 ^ (p - 1) - 1 = 0 in ℕ
isWieferichPrimeBase_one_iff Every prime is a Wieferich prime to base 1
IsWieferichPrime Wieferich prime A Wieferich prime is a prime $p$ with $2^{p-1} \equiv 1 \pmod{p^2}$, i.e. a Wieferich prime to base $2$
IsMirimanoffPrime Mirimanoff prime A Mirimanoff prime is a prime $p$ with $3^{p-1} \equiv 1 \pmod{p^2}$, i.e. a Wieferich prime to base $3$
Filter.cofinite_hasBasis_Ici Documented declaration: cofinite_hasBasis_Ici
UnimodularOn f is unimodular on (a, ∞) if it increases until some point M > a and decreases thereafter
FormalConjecturesForMathlib.Probability.Finset.exists_eq_zero_of_sum_lt_card Finite probabilistic method (integer form)
FormalConjecturesForMathlib.Probability.Finset.exists_eq_zero_of_real_sum_lt_card Finite probabilistic method (real form)
ContinuousCohomology.tateShafarevich The Tate–Shafarevich group of a Galois module
Bialgebra.cohomologyRing The cohomology ring H^*(A, k) = ⨁ n, Ext^n_A(k, k) of the bialgebra A over k, where k is the trivial A-module through the counit and Ext is taken in ModuleCat A
Bialgebra.cohomology The cohomology H^*(A, M) = ⨁ n, Ext^n_A(k, M) of the bialgebra A over k with coefficients in the A-module M
Bialgebra.cohomologyRing_of_mul_of The product on cohomologyRing k A is the Yoneda composition of extensions
Bialgebra.cohomologyRing_algebraMap_apply A scalar c : k acts on cohomologyRing k A as c times the identity in degree 0
Bialgebra.nontrivial_cohomologyRing The cohomology ring is not the zero ring: its degree-zero part contains the identity of the trivial module
Bialgebra.cohomology_of_smul_of The action of cohomologyRing k A on cohomology k A M is the Yoneda composition of extensions
Bialgebra.trivialModuleCat The trivial module of the bialgebra A over R: the ring R on which A acts through the counit A →ₐ[R] R, as an object of ModuleCat A
Bialgebra.trivialModuleCat_smul The bialgebra A acts on its trivial module through the counit
Bialgebra.trivialModuleCat_smul_base Documented declaration: trivialModuleCat_smul_base
Bialgebra.trivialModuleCatLinearEquiv The trivial module of a bialgebra over a field k is k itself as a k-module; in particular it is one-dimensional
Module.IsSmallCohenMacaulay Documented declaration: IsSmallCohenMacaulay
Module.IsBalancedBigCohenMacaulay Documented declaration: IsBalancedBigCohenMacaulay
Module.IsSmallCohenMacaulay.nontrivial A small Cohen-Macaulay module is nonzero: a regular sequence on M is one for which M is not killed by the ideal it generates
Module.IsBalancedBigCohenMacaulay.nontrivial A balanced big Cohen-Macaulay module is nonzero: on the zero module the maximal ideal acts by the identity
Module.smul_top_ne_top_of_isRegular Documented declaration: smul_top_ne_top_of_isRegular
Module.isSmallCohenMacaulay_self Over a field K, the module K is a small Cohen-Macaulay module
RightIdeal.mul_mem_right A right ideal is closed under multiplication on the right
RightIdeal.toIdeal A right ideal in a commutative semiring, regarded as an ideal
RightIdeal._root_.Ideal.toRightIdeal An ideal in a commutative semiring, regarded as a right ideal
RightIdeal.orderIsoIdeal Over a commutative semiring, right ideals and ideals are order isomorphic
RightIdeal A right ideal in a semiring R is an additive submonoid closed under multiplication on the right
RightIdeal.IsMaximal A right ideal is maximal if it is maximal among the proper right ideals
Module.intersectionMultiplicity The intersection multiplicity of two modules M and N over a commutative ring R
Module.intersectionMultiplicity_of_projective If N is projective, only Tor₀(M, N) = M ⊗ N contributes to χ(M, N)
Module.intersectionMultiplicity_eq_zero_of_subsingleton If M is the zero module, then χ(M, N) = 0: the functor M ⊗ - is zero, so all its left derived functors vanish
Module.intersectionMultiplicity_self_self Normalisation: over a field k, χ(k, k) = 1
MvPolynomial.hilbertFunction The Hilbert function of a homogeneous ideal I of $k[x_1, \dots, x_n]$, sending d to $\dim_k I_d$
MvPolynomial.IsMonomialIdeal An ideal of a polynomial ring is a monomial ideal if it is generated by monomials
MvPolynomial.IsLexIdeal A lex ideal is a monomial ideal L such that for monomials u, v of the same degree with u larger than v in the lexicographic order, v ∈ L implies u ∈ L
MvPolynomial.purePowersIdeal The ideal generated by the pure powers $x_1^{d_1}, \dots, x_c^{d_c}$
MvPolynomial.hilbertFunction_bot The zero ideal has Hilbert function zero
MvPolynomial.purePowersIdeal_zero For c = 0 there are no pure powers, so purePowersIdeal is the zero ideal
MvPolynomial.isLexIdeal_top The unit ideal is a lex ideal
IsRightNoetherianRing A semiring is right Noetherian if it is Noetherian as a right module over itself
IsLocalRing.IsSystemOfParameters A system of parameters of a local ring R is a list of ringKrullDim R elements of R generating an ideal whose radical is the maximal ideal
IsLocalRing.IsSystemOfParameters.mem_maximalIdeal The elements of a system of parameters lie in the maximal ideal
IsLocalRing.isSystemOfParameters_nil Over a local ring of Krull dimension zero, the empty list is a system of parameters
ValuationSubring.centerOn The centre of a valuation subring 𝒪 of F on a k-subalgebra A of F contained in 𝒪: the prime ideal 𝔪_𝒪 ∩ A of A
ValuationSubring.mem_centerOn_iff The centre of 𝒪 on A consists of the elements of A of valuation less than one
ValuationSubring.HasLocalUniformization Documented declaration: HasLocalUniformization
ValuationSubring.hasLocalUniformization_of_isRegularRing A regular affine model of F over k inside 𝒪 uniformizes 𝒪
OrdinalCardinalRamsey This proposition asserts the ordinal Ramsey property α → (β, c)²
Lean.Elab.ContextInfo.runMetaMInCommandElabM A copy of ContextInfo.runMetaM that keeps the messages and passes in the same filename
Lean.Elab.Command.Linter.runTermLinter A linter that applies to terms
IsAbsoluteNeighborhoodRetract Documented declaration: IsAbsoluteNeighborhoodRetract
not_summable_subtype_iff_tendsto_sum_indicator For a nonnegative f, the sum over a subset A fails to be summable exactly when the partial sums of f restricted to A diverge
TopologicalSpace.IsNetwork A family N of subsets of a topological space X is a network for X if every open subset of X is a union of members of N
TopologicalSpace.IsNetwork.preimage The preimage of a network under an inducing map is a network
TopologicalSpace.density The density d(X) of a topological space X: the least cardinality of a dense subset, plus ℵ₀
TopologicalSpace.networkWeight The network weight nw(X) of a topological space X: the least cardinality of a network, plus ℵ₀
TopologicalSpace.isNetwork_range_sInter The sets ⋂₀ {U | IsOpen U ∧ x ∈ U}, for x : X, form a network of X
TopologicalSpace.exists_dense_mk_add_aleph0_eq_density The density is attained by some dense subset
TopologicalSpace.exists_isNetwork_mk_add_aleph0_eq_networkWeight The network weight is attained by some network
TopologicalSpace._root_.Topology.IsInducing.networkWeight_le The network weight of a subspace is at most that of the ambient space
Homeomorph.OfDiscrete A bijection between discrete topological spaces induces a homeomorphism
IsHomeomorph.equiv_of_discreteTopology A bijection between discrete topological spaces is a homeomorphism
HasGδSingletons A space where all singletons are Gδ sets
HasGδSingletons.of_t1Space_firstCountableTopology Singletons are Gδ in first-countable T₁ spaces
HomogeneousSpace A space is homogeneous if its homeomorphism group acts transitively
HasLebesgueCoveringDimensionLE Documented declaration: HasLebesgueCoveringDimensionLE
HasLebesgueCoveringDimensionEq A topological space X has Lebesgue covering dimension exactly n if it has covering dimension at most n but not at most n - 1
Metric.gromovProduct The Gromov product of y and z with respect to x
Metric.IsHyperbolicWith A pseudometric space is δ-hyperbolic if for all w x y z the four-point condition min (gromovProduct w x y) (gromovProduct w y z) - δ ≤ gromovProduct w x z holds
Metric.isHyperbolicWith_of_forall_dist_le A pseudometric space with all distances bounded above by k is k-hyperbolic
Metric.isHyperbolicWith_diam_univ A bounded space is δ-hyperbolic with respect to its diameter
Metric.IsHyperbolic A pseudometric space is hyperbolic if it is δ-hyperbolic for some real constant δ
Metric.IsSeparated' A set s is ≥ ε-separated if its elements are pairwise at distance greater or equal to ε from each other
AMSLinter.test_theorem A highly non-trivial theorem
AMSLinter.test_theorem_with_hypothesis A highly non-trivial theorem with a helpful hypothesis
AMSLinter.test_theorem_with_hypotheses A highly non-trivial theorem with two helpful hypotheses
AMSLinter.test_theorem_with_docstring A highly non-trivial theorem with a helpful hypothesis
CategoryAnswerLinter.flagged_solved_answer_sorry A solved problem should not leave its answer unfilled
CategoryAnswerLinter.flagged_ascribed_answer_sorry The sorry is still a placeholder when it carries a type ascription
CategoryAnswerLinter.flagged_let_answer_sorry An answer bound by a let is flagged too
CategoryAnswerLinter.not_flagged_explicit_answer A solved problem with an explicit answer is fine
CategoryAnswerLinter.not_flagged_open_answer_sorry An open problem may leave its answer unfilled
CategoryLinter.test_theorem A highly non-trivial theorem
CategoryLinter.test_theorem_with_hypothesis A highly non-trivial theorem with a helpful hypothesis
CategoryLinter.test_theorem_with_hypotheses A highly non-trivial theorem with two helpful hypotheses
CategoryLinter.test_theorem_with_docstring A highly non-trivial theorem with a helpful hypothesis
CategoryLinter.test_theorem_match A highly non-trivial theorem proved by cases
FormalProofLinter.an_open_assumption An assumption that is still open, which is the ordinary case
FormalProofLinter.a_proved_assumption An assumption that has since been proved
Google.AnswerSetting A type that captures the current setting for the answer() elaborator
Google.findAnswerExpr Find the first subexpression carrying the answer annotation, returning the inner (unwrapped) expression if found
Google.findAnswerExprs Collect all subexpressions carrying the answer annotation, returning the inner (unwrapped) expressions
Google.mkCanonicalSorryAnnotation Construct a canonical sorryAx application with a stable, module-independent tag
Google.answerElab Indicates where the answer is in a problem statement
Google.AnswerInfo An answer: a term, and the context in which it was elaborated
Google.AnswerInfo.format Print an answer
Google.getAnswers Find answers by inspecting an InfoTree
ProblemAttributes.FormalProofKind The type of formal proof that exists for a problem
ProblemAttributes.problemStatus.toName Convert from a syntax node to a name
ProblemAttributes.Category A type to capture the various types of statements that appear in our Lean files
ProblemAttributes.CategoryTag TODO(lezeau): do we eventually want to account for the problem's source?
ProblemAttributes.FormalProofTag A tag recording the existence and location of a formal proof for a declaration
ProblemAttributes.Syntax.toCategory Convert from a syntax node to a term of type Category and annotate the syntax with the corresponding name's docstring
ProblemAttributes.toCategorySyntax Extract the category attributes from a declaration's modifiers
ProblemAttributes.toCategories Extract the categories from a declaration's modifiers
ProblemAttributes.addFormalProofAttribute Records the existence and location of a formal proof for a statement
ProblemAttributes.Syntax.toSubjects Converts a syntax node to an array of AMS subjects
ProblemAttributes.splitByFun Split an array into preimages of a function
ProblemAttributes.getFormalProofTag Get the formal proof tag for a given declaration, if any
ProblemAttributes.getProofConditions Get the unproven hypotheses a given declaration's formal proof assumes
ProblemAttributes.verifyCategoryCounts Verify that the list of problems contains the expected number of problems for each category
AMSLinter.toAMS Checks if a command has the AMS attribute
AMSLinter.AMSLinter The problem category linter checks that every theorem/lemma/example has been given an AMS attribute
AnswerLinter.contains_early_args Helper to check for arguments (h : True)
AnswerLinter.stars_with_answer_sorry_iff Helper to check for answer(sorry) ↔
AnswerLinter.answerLinter The actual linter object
AnswerLinter.flagged_by_linter An exampe of what we want to lint against
AnswerLinter.not_flagged_no_answer_sorry An non-exampe: we want don't to lint against this case
AnswerLinter.not_flagged_no_arguments An non-exampe: we want don't to lint against this case
AnswerLinter.not_flagged_non_prop_answer An non-exampe: here answer(sorry) is not a Prop, and not the entire left hand side of the iff
CategoryAnswerLinter.containsSorry Whether stx contains a sorry anywhere
CategoryAnswerLinter.answerSorries The answer(..) nodes in stx whose answer is still a sorry
CategoryAnswerLinter.categoryNeedsAnswer Whether the given category is research solved
CategoryAnswerLinter.categoryAnswerLinter The linter checking that solved statements do not use answer(sorry)
CategoryDocstringLinter.categoryNeedsDocstring Whether the given category requires a docstring
CategoryDocstringLinter.hasDocstring Whether the declaration modifiers contain a docstring
CategoryDocstringLinter.categoryDocstringLinter The linter checking for docstrings on research statements
CategoryDocstringLinter.not_flagged_with_docstring A documented open problem statement should not be flagged
CategoryLinter.toCategory Checks if a command has the category attribute
CategoryLinter.checkNotOpenIfSorryFree Warns when a problem categorised as research open turns out to have a sorry-free proof
CategoryLinter.checkExactlyOneCategory Warns unless a carries exactly one category attribute
CategoryLinter.categoryLinter The problem category linter checks that every theorem/lemma/example has been given a problem category attribute
CopyrightLinter.correctCopyrightPrefix The part of the expected copyright before the year
CopyrightLinter.correctCopyrightSuffix The part of the expected copyright after the year
CopyrightLinter.hasCorrectCopyright Check whether a file, given as a String, is prefixed with the correct copyright header
CopyrightLinter.correctCopyrightHeader The current correct copyright header
CopyrightLinter.copyrightLinter The copyright linter ensures that every file has the right copyright header
ExistsImplicationLinter.forallToAnd Changes and expression of the form ∀ (h1 : Prop1) (h2 : Prop2) ..., Propn to `Prop1 ∧ Prop2 ∧
ExistsImplicationLinter.checkExistsArrow Checks if an expression contains the pattern ∃ x, P x → Q
ExistsImplicationLinter.existsImplicationLinter The existsImplicationLinter detects expressions of the form ∃ a, P a → Q and flags them to the user since those are rarely correct
FormalProofLinter.isProved Whether declName has a proof term with no sorry in it, waiting on the declaration to finish elaborating
FormalProofLinter.getFormalProofTagsFor Return every formal-proof tag attached to declName
FormalProofLinter.checkAssumptionsStillOpen Warns when a hypothesis assumed by a conditional formal_proof has since been proved, so the proof may no longer be conditional on it
FormalProofLinter.checkProofIsSorryIfFormalProof Warns when a statement with a formal_proof annotation has a proof that is not exactly sorry or by sorry, unless it is conditional
FormalProofLinter.formalProofLinter Checks the assumptions named by a conditional formal_proof and ensures formal_proof statements are sorry
ImportLinter.ImportRef An import of a parsed header, retaining its syntax node
ImportLinter.ImportRef.getIdent Extracts the module identifier from an ImportRef
ImportLinter.headerToImportRefs Collects every import of a parsed header, with its modifiers and syntax
ImportLinter.checkImports Checks a parsed header against the Formal Conjectures import rules
ImportLinter.importLinter The import linter ensures that: - Files in FormalConjectures do not import Mathlib, Mathlib.*, FormalConjecturesForMathlib, or FormalConjecturesForMathlib.* directly
LatexDocstringLinter.latexDocstringLinter The linter checking markup in docstrings
LatexDocstringLinter.flagged_with_bracket_math This is a bad docstring with \[ math \]
LatexDocstringLinter.flagged_with_paren_math This is a bad docstring with $ math $
LatexDocstringLinter.not_flagged_with_good_math This is a good docstring with $$ math $$
LatexDocstringLinter.flagged_with_legacy_url_link This is a bad link: [example][https://example.com]
LatexDocstringLinter.not_flagged_with_inline_url_link This is a good link: example
ModuleDocstringLinter.moduleDocstringLinter The module docstring linter checks two things: - A file that has no `/-!
NamespaceLinter.namespaceLinter The namespace linter checks that every declaration is inside a namespace
StubLinter.hasSorry Checks whether a syntax node contains sorry, admit, or sorryAx
StubLinter.checkDecl Checks a declaration command for stubs, placeholder definitions, and axioms
StubLinter.stubLinter The stub linter checks that no opaque, axiom, or `def