Glossary: P

Definitions in this project whose names start with P. The whole glossary has the other sections.

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P def

The class P is the set of decision problems decidable in polynomial time by a deterministic Turing machine. · ComplexityTheory.P

P def

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.P

P def

Let $P(n, k)$ be the largest prime factor of $\binom{n}{k}$. · Erdos683.P

P def

P n is the largest prime factor of n (and 0 if n ≤ 1). · Erdos700.P

P def

$P(i)$ is the $i$-th prime, 1-indexed. · OeisA167918.P

P def

The finite polynomial approximation $\prod_{k=1}^n (1 - X^k / k!)$. · OeisA185895.P

p def

A prime $p$ is in class $1$ if the only prime divisors of $p+1$ are $2$ or $3$. In general, a prime $p$ is in class $r$ if every prime factor of $p+1$ is in some class $\leq r-1$, with equality for at least one prime factor. Let $p_r$ is … · Erdos1055.p

p def

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.p

p def

Let $z_i$ be an infinite sequence of complex numbers such that $|z_i| = 1$ for all $i \geq 1$. For $n \geq 1$ let $p_n(z) = \prod_{i \leq n} (z - z_i)$. · Erdos119.p

p def

Let $p_n$ be the smallest prime $\equiv 1\pmod{n}$. · Erdos456.p

p abbrev

The nth prime, indexed from zero: p 0 = 2, p 1 = 3, ... · Erdos682.p

p def

Proposition that for a specific $n$ an $A$ with the above defined condition and cardinality $k$ exists. · Erdos888.p

p def

def p (α : ℝ) : ℝ² · MovingSofa.GerversSofa.p

p def

The least prime $p > \sqrt{n!}$ such that $(p - 1) \mid n!$ and $q = \frac{n!}{p - 1} + 1$ is prime. · OeisA55487.p

Packing structure

A structure representing a packing of n isometric embeddings of a set s inside a (presumably larger) set S. · SquarePacking.Packing

PairwiseCoprime def

Predicate for pairwise coprimality of three integers. Requires all three input values to be pairwise coprime to each other. · Erdos123.PairwiseCoprime

PairwiseDistancesDistinct def

A finite set of points has all pairwise distances distinct. · Erdos1088.PairwiseDistancesDistinct

parseTuring def

parseTuring tmStringDescription outputs the Expr corresponding to the Turing Machine described by the string tmStringDescription. · parseTuring

partialDensity abbrev

Given a set S and an element b in an order β, where all intervals bounded above are finite, we define the partial density of S (relative to a set A) to be the proportion of elements in {x ∈ A | x < b} that lie in S ∩ A. · Set.partialDensity

partialLcm def

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.partialLcm

partialSum def

The partial sum $S_N(\alpha) = \sum_{k < N} f(\{\alpha n_k\})$. · Erdos995.partialSum

partiteExtremalNumber def

The extremal number forbidding a complete balanced multipartite uniform hypergraph. · Hypergraph.partiteExtremalNumber

partQuot def

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 $0$. The integer part $c_0$ is not part of the sequence. The expansion of an irrational number never … · Real.partQuot

parts def

The n-element subsets of {1, …, 2n}. · Erdos36.parts

path def

The path number of a graph: The number of vertices of a largest induced path of the graph. · SimpleGraph.path

pathCoverNumber def

Minimum size of a path cover of G. · SimpleGraph.pathCoverNumber

PebbleDistribution def

A Pebble distribution is an assignment of zero or more pebbles to each of the vertices. · PebblingNumberConjecture.PebbleDistribution

PebblePath inductive

A pebble path is a series of pebbling moves. · PebblingNumberConjecture.PebblePath

PebblingNumber def

The pebbling number of a graph G, is the lowest natural number n that satisfies the following condition: Given any target or 'root' vertex in the graph and any initial pebbles distribution with n pebbles on the graph, another pebble … · PebblingNumberConjecture.PebblingNumber

pellNumber def

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

pendantVertices def

The set of pendant vertices (leaves) of G: vertices of degree 1. · SimpleGraph.pendantVertices

pentagonal def

$p_5(y) = \frac{3y^2 - y}{2}$ is the $y$-th pentagonal number. · OeisA160324.pentagonal

perfectDeficiency def

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.perfectDeficiency

PerfectFor def

A positive integer $n$ is $(m,k)$-perfect if $\sigma^m(n) = kn$ where $\sigma^m$ is the $m$-th iterate of $σ$. · Superperfect.PerfectFor

perfectMatchingCount def

The number of perfect matchings of G. · LovaszPlummerConjecture.perfectMatchingCount

PerfectRuler def

An $N$-perfect ruler is a finite subset $A \subseteq \mathbb{N}$ (the marks), such that each positive integer $k \leq N$ can be measured, that is, expressed as a difference $k = a_1 - a_0$ with $a_0, a_1 \in A$. The set $A$ is then also … · Erdos170.PerfectRuler

PerfectRulersLengthN def

We define the set of all $N$-perfect rulers $A$ of length $N$, i.e. subsets $A \subseteq \{0, \dots, N\}$, s.t. $A$ is $N$-perfect. This is also called a restricted difference basis w.r.t. $N$. · Erdos170.PerfectRulersLengthN

periodOfPowerMod def

The length of the minimal positive period of the sequence $k^p \pmod n$. $a_p(n) = \min \{ T \in \mathbb{N}^+ \mid \forall k \in \mathbb{N}, (k+T)^p \equiv k^p \pmod n \}$. · OeisA282779.periodOfPowerMod

peripheryMaxDegree def

The maximum original-graph degree among the peripheral vertices. · WrittenOnTheWallII.GraphConjecture172.peripheryMaxDegree

permuteConfig def

Permute the parties of a configuration. · OpenQuantumProblem35.permuteConfig

permuteState def

Permute the parties of a state vector. · OpenQuantumProblem35.permuteState

PetersenGraph abbrev

Petersen Graph on 10 vertices. · WrittenOnTheWallII.Test.PetersenGraph

phaseMatrix def

The raw phase-parametrized Hadamard matrix. The cases $\zeta = 1$ and $\zeta = i$ give the $X$ and $Y$ bases. · OpenQuantumProblem13.Qubit.phaseMatrix

phaseU def

The bundled qubit basis associated to a unit-modulus phase $\zeta$. · OpenQuantumProblem13.Qubit.phaseU

phi def

The golden ratio $\phi = (1 + \sqrt{5})/2$. · OeisA105565.phi

phiAvg def

The average $\frac{1}{N}\sum_{k\leq N}\frac{\phi_A(k)}{n_k}$. · Erdos1000.phiAvg

phiSeq def

Given an infinite sequence of positive integers $A = \{n_1 < n_2 < \cdots\}$, phiSeq n k is $\phi_A(k)$: the number of $1\leq m\leq n_k$ such that $\frac{m}{n_k}$ cannot be written as $\frac{b}{n_j}$ for any integer $b$ and any $1\leq … · Erdos1000.phiSeq

PillaiPrimes abbrev

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.PillaiPrimes

pinnedDistinctDistances def

The pinned distance function $f(n)$. For a set $A$ of $n$ points in $\mathbb{R}^2$, take the largest number of distinct distances from a point $x \in A$ to the other points of $A$. Then $f(n)$ is the least such value over all such sets $A$. · Erdos604.pinnedDistinctDistances

piRestricted def

$\pi_n$ counts how many of $S_1,S_2,\dots,S_n$ are themselves prime, i.e. $\pi_n$ as used on MathOverflow (compare to $\pi_n$ in [Me18, Conjecture 3.3], which instead ranges over $S_2,S_4,\dots,S_{2n}$). · MathOverflow434111.piRestricted

pisotCantorSet def

The set $C(\alpha)$ of Problem 10.61. For $\alpha > 2$ it is a Cantor set of Hausdorff dimension $\log 2 / \log \alpha < 1$, normalised so that $0$ is its least and $1$ its greatest element. · Bugeaud61.pisotCantorSet

Player inductive

An arbitrary two elements type indexing the players in the Catch-Up game. · CatchUp.Player

pmSumList def

Perfect-matching sum on a list: run pmSumListAux with fuel = L.length. · MonochromaticQuantumGraph.pmSumList

pmSumListAux def

Auxiliary perfect-matching sum on a vertex list, using a fuel parameter n for termination. · MonochromaticQuantumGraph.pmSumListAux

pmSumN def

The perfect-matching sum for $K_N$: use the canonical ordered vertex list vertices N. · MonochromaticQuantumGraph.pmSumN

Point abbrev

A point of Euclidean three-space. · AtiyahSutcliffe.Point

pointMultiplicity def

The number of lines from A that pass through the point p. · Erdos209.pointMultiplicity

pointPolynomial def

The polynomial associated to point i in a finite configuration x. · AtiyahSutcliffe.pointPolynomial

pointsOn def

The number of points of P that lie on L. · Erdos102.pointsOn

poissonBracket def

The canonical Poisson bracket on MvPolynomial (σ ⊕ σ) R, given by $\{f, g\} = \sum_{i} (\partial_{x_i} f \cdot \partial_{y_i} g - \partial_{y_i} f \cdot \partial_{x_i} g)$ where $x_i$ (resp. $y_i$) is the variable indexed by Sum.inl i … · MvPolynomial.poissonBracket

PoissonConjectureFor def

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. An endomorphism is an automorphism iff it is bijective, since the inverse … · Arxiv.«math.0608009».PoissonConjectureFor

polygonalNumber def

$p_k(x) = \frac{(k-2)x(x-1)}{2} + x$ is the $x$-th $k$-gonal number. · OeisA160324.polygonalNumber

polynomial def

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.polynomial

polyP def

def polyP (n : ℕ) : Polynomial ℚ · OeisA115257.polyP

polyQ def

def polyQ (n : ℕ) : Polynomial ℚ · OeisA115257.polyQ

polyToLaurent def

The natural inclusion F[t] →+* F((t⁻¹)). The field F((t⁻¹)) is modelled by the Laurent series field F⸨X⸩, whose variable X plays the role of t⁻¹, so the polynomial variable t is sent to X⁻¹. · Margulis.polyToLaurent

positive_lower_density def

A set of natural numbers has positive lower density. · Erdos16.positive_lower_density

positiveCount def

The count of positive terms among $a(1)$, ..., $a(N)$. · OeisA117027.positiveCount

possible_f_values def

The set of all possible values k for which there exists a set of n points satisfying the hasMinEquidistantProperty k. The function f(n) will be the supremum of this set. · Erdos92.possible_f_values

possibleUnitCircleCounts def

The set of unit-circle counts attained by configurations of n points in the plane. · Erdos104.possibleUnitCircleCounts

powCorrectionFactor def

Artin's conjecture on $S(a)$ when $a = b^m$ is a power, where $m$ is odd and maximal, requires a correction factor to multiply ArtinConstant and is given by $$\prod_{p \mid m} \frac{p(p - 2)}{p^2 - p - 1}.$$ · ArtinPrimitiveRootsConjecture.powCorrectionFactor

Powerfree def

def Powerfree (k : ℕ) (m : M) : Prop · Powerfree

Powerful abbrev

A Powerful number is a natural number $n$ where for every prime divisor $p$, $p^2$ divides $n$. Powerful numbers are also known as "squareful", "square-full", or "$2$-full". · Nat.Powerful

powerSum def

The $k$th power sum of a finite sequence of complex numbers. · Erdos519.powerSum

powerSum def

The power sum $S_k(m)=\sum_{i=1}^{m-1} i^k$. · ErdosMoser.powerSum

pPoly def

The polynomial $pPoly(X, Y, Z) = (1 + X + Y + Z)^{2n} (1 + X + Y - Z)^n (1 + X - Y + Z)^n$. We identify $X_0, X_1, X_2$ with $X, Y, Z$. · OeisA2897.pPoly

practicalH def

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

PreservesRationality def

A real function preserves rationality. · Erdos226.PreservesRationality

prime def

$\mathrm{prime}(k)$ is the $k$-th prime number ($\mathrm{prime}(1) = 2$). · OeisA182126.prime

primeArithmeticProgressions def

The set of arithmetic progressions of primes · Erdos219.primeArithmeticProgressions

primeCandidates def

The finite set in which the least prime strictly above n is sought. · OeisA67836.primeCandidates

primeCount_Ioc_mul_const def

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 $C$. · Erdos1138.primeCount_Ioc_mul_const

primeCountInInterval def

The number of primes $p < N$ for which the normalized coefficient belongs to $[a,b]$. · SatoTateConjecture.primeCountInInterval

primeDensityRatio def

The density ratio of set $A$ up to $n$ relative to the prime counting function $\pi(n)$. · Erdos428.primeDensityRatio

primeFactorSplice def

Concatenate the prime factors of n in nondecreasing order, retaining multiplicity. · OeisA37274.primeFactorSplice

primeGap def

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.primeGap

primeGap def

The prime gap: the difference between the $n+1$-th and $n$-th prime. · primeGap

primePrefix def

The prime factors q₁ < ⋯ < qⱼ of N which are at most q. · Erdos534.primePrefix

primesAtTop def

atTop restricted to prime numbers. · Green27.primesAtTop

primeSet def

The set of primes p such that Nat.floor (α * p) is also prime. · Erdos972.primeSet

primesInAPCount def

$\pi(x; q, a)$: the number of primes $p \le x$ with $p \equiv a \pmod q$. · ElliottHalberstamConjecture.primesInAPCount

primeTupleCounting def

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.Nat.primeTupleCounting

primitives def

The set of primitive elements of S. · Set.primitives

PrimitiveWeird def

Primitive weird numbers are weird numbers such that no proper divisor of $n$ are weird. · Erdos470.PrimitiveWeird

prob def

The probability that a uniformly random degree-n polynomial with $\pm 1$ coefficients satisfies P. · Erdos525.prob

ProbabilityMeasure def

def UnitAddCircle.ProbabilityMeasure : ProbabilityMeasure 𝕋 · Arxiv.id2303_01089.UnitAddCircle.ProbabilityMeasure

problems def

def problems : List Name · Subsets.FC100OpenSet1.problems

problems def

def problems : List Name · Subsets.FC100SolvedSet1.problems

ProblemStatus inductive

inductive ProblemStatus /-- Indicates that a mathematical problem is still open. -/ · ProblemAttributes.ProblemStatus

prodFactorMinus def

The factor $\prod_{k=1}^{2m} (2mn - k)$ · OeisA103885.prodFactorMinus

prodFactorPlus def

The factor $\prod_{k=1}^{2m} (2mn + k)$ · OeisA103885.prodFactorPlus

ProdFree def

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.ProdFree

prodMultiset def

A version in [Er61] by Erdős is product instead of sum, which is false. Counterexample (by Steinerberger): consider $k = 3$ and let $A = \{1, \zeta_6, \zeta_6^2, \zeta_6^4\}$ and $B = \{1, \zeta_6^2, \zeta_6^3, \zeta_6^4\}$. · Erdos494.prodMultiset

product def

Product of the first $n$ terms of Chua's sequence. · OeisA167604.product

productIndices def

The indices $k = 1$ to $2m$, used in the product · OeisA103885.productIndices

productSet def

The product $A \times B$, represented by consecutive coordinate blocks. · SpectralSetProduct.productSet

productsOfSubsets def

The set of products of subsets of $\{1, \dots, n\}$. · OeisA60957.productsOfSubsets

ProjectiveSpace def

The projective space over a scheme S, with homogeneous coordinates indexed by n · ProjectiveSpace

PromislowGroup abbrev

The Promislow group ⟨ a, b | b⁻¹a²ba², a⁻¹b²ab² ⟩ · Kaplansky.PromislowGroup

ProOrderable def

ProOrderable G means every bi-invariant partial order extends to a bi-invariant linear order. · Kourovka.«1.35c».ProOrderable

Property def

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.Property

Property25 def

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.Property25

propertyBWitnesses def

Subsets of the incident vertices witnessing Property B. · Finset.propertyBWitnesses

properUnitaryDivisors def

A proper unitary divisor of $n$ is a divisor $d$ of $n$ such that $d$ is coprime to $n/d$, and $d < n$. · Erdos1052.properUnitaryDivisors

proportion def

The proportion appearing in erdos_235, for the primorial N k. · Erdos235.proportion

proportionCoverable def

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.proportionCoverable

proves def

def proves (L : NormalModalLogic) (φ : Formula) · Arxiv.«1308.0994».proves

purePowersIdeal def

The ideal generated by the pure powers $x_1^{d_1}, \dots, x_c^{d_c}$. · MvPolynomial.purePowersIdeal

pyjamaSet def

The pyjama set is the set of points in the complex plane whose real part is within $\varepsilon$ of an integer. · Green41.pyjamaSet

pythagorasBounds def

The set of p such that every sum of squares in R is a sum of p squares. Its least element, when it exists, is the Pythagoras number of R. · pythagorasBounds

P₁ def

A set A : Set ℕ is said to have property P₁ if for any nonconstant sequence f : A → {-1, 1}, one can always select a finite, nonempty subset S ⊆ A \ {0} such that ∑ n ∈ S, fₙ / n = 0. This is defined in [Sa82b]. · Erdos318.P₁