441.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 6 contain sorry, 0 declared as axioms

Erdős Problem 441

References:

erdosproblems.com/441
[Er51b] P. Erdős, Problem. Mat. Lapok (1951), 233.
[Er65] Erdős, P., Extremal problems in number theory. Proc. Sympos. Pure Math., Vol. VIII (1965), 181-189.
[Er73] Erdős, P., Problems and results on combinatorial number theory. A survey of combinatorial theory (Proc. Internat. Sympos., Colorado State Univ., Fort Collins, Colo., 1971) (1973), 117-138.
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
[Er98] Erdős, Paul, Some of my new and almost new problems and results in combinatorial number theory. Number theory (Eger, 1996) (1998), 169-180.
[Ch72b] Choi, S. L. G., The largest subset in $[1,n]$ whose integers have pairwise L.C.M. not exceeding $n$. Mathematika (1972), 221-230.
[Ch98] Chen, Yong-Gao, Sequences with bounded l.c.m. of each pair of terms. Acta Arith. (1998), 71-95.
[DaCh06] Dai, Li-Xia and Chen, Yong-Gao, Sequences with bounded l.c.m. of each pair of terms. II. Acta Arith. (2006), 315-326.
[ChDa07] Chen, Yong-Gao and Dai, Li-Xia, Sequences with bounded l.c.m. of each pair of terms. III. Acta Arith. (2007), 125-133.
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.

Imports (1)

FormalConjecturesUtil

def Erdos441.g g N is the size of the largest A ⊆ {1, …, N} such that lcm(a, b) ≤ N for all a, b ∈ A line 55 source
Source documentation
g N is the size of the largest A ⊆ {1, …, N} such that lcm(a, b) ≤ N for all a, b ∈ A.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: SupSet.sSup, Set.ofPred, Exists, Finset, And, LE.le, Finset.Icc, OfNat.ofNat, Membership.mem, Nat.lcm, Eq, Finset.card
10 instances, coercions and auxiliary constants left out.
```
noncomputable def g (N : ℕ) : ℕ :=
…
```
def Erdos441.erdosConstruction Erdős' construction: all integers in $[1,(N/2)^{1/2}]$ together with all even integers in $[(N/2)^{1/2},(2N)^{1/2}]$ line 63 source
Source documentation
Erdős' construction: all integers in $[1,(N/2)^{1/2}]$ together with all even integers in $[(N/2)^{1/2},(2N)^{1/2}]$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Finset
Proof uses: Finset.filter, Or, LE.le, HMul.hMul, OfNat.ofNat, HPow.hPow, And, Dvd.dvd, Finset.Icc
15 instances, coercions and auxiliary constants left out.
```
def erdosConstruction (N : ℕ) : Finset ℕ :=
…
```
theorem Erdos441.erdos_441 sorry line 90 source
depends on sorry what this means
Statement uses: Iff, False, Nat, LE.le, OfNat.ofNat, Eq, Erdos441.g, Finset.card, Erdos441.erdosConstruction
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem Erdos441.erdos_441.variants.chen_dai_infinitely_often Chen and Dai [ChDa07] proved that, infinitely often, Erdős' construction is not optimal: for infinitely many sorry line 101 source
Source documentation
Chen and Dai [ChDa07] proved that, infinitely often, Erdős' construction is not optimal: for infinitely many $N$, $g(N) \geq \lvert B\rvert+t-2$, where $B$ is the construction and $t\geq 0$ is such that the $t$-fold iterated logarithm of $N$ is in $[0,1)$. (erdosproblems.com omits the $-2$; [ChDa07] proves the bound with it.)

depends on sorry what this means
Statement uses: Filter.Frequently, Nat, LE.le, HAdd.hAdd, Finset.card, Erdos441.erdosConstruction, Real.iteratedLog, Real, Erdos441.g, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
14 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_441.variants.chen_dai_infinitely_often :
    ∃ᶠ N : ℕ in atTop, (erdosConstruction N).card + Real.iteratedLog N ≤ g N + 2 := by
…
```
Mentions: Erdos441.erdosConstruction, Real.iteratedLog, Erdos441.g
theorem Erdos441.erdos_441.variants.lower_bound Erdős' construction proves that $g(N) \geq \left(\tfrac{9}{8}N\right)^{1/2}+O(1)$ sorry line 107 source
Source documentation
Erdős' construction proves that $g(N) \geq \left(\tfrac{9}{8}N\right)^{1/2}+O(1)$.

depends on sorry what this means
Statement uses: Exists, Real, Nat, LE.le, HSub.hSub, Real.sqrt, HMul.hMul, HDiv.hDiv, OfNat.ofNat, Erdos441.g
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
28 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_441.variants.lower_bound :
    ∃ C : ℝ, ∀ N : ℕ, √(9 / 8 * N) - C ≤ g N := by
…
```
Mentions: Erdos441.g
theorem Erdos441.erdos_441.variants.erdos_upper_bound Erdős [Er51b] proved $g(N) \leq (4N)^{1/2}+O(1)$ sorry line 113 source
Source documentation
Erdős [Er51b] proved $g(N) \leq (4N)^{1/2}+O(1)$.

depends on sorry what this means
Statement uses: Exists, Real, Nat, LE.le, Erdos441.g, HAdd.hAdd, Real.sqrt, HMul.hMul, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_441.variants.erdos_upper_bound :
    ∃ C : ℝ, ∀ N : ℕ, (g N : ℝ) ≤ √(4 * N) + C := by
…
```
Mentions: Erdos441.g
theorem Erdos441.erdos_441.variants.chen Chen [Ch98] established the asymptotic $g(N) \sim \left(\tfrac{9}{8}N\right)^{1/2}$ sorry line 119 source
Source documentation
Chen [Ch98] established the asymptotic $g(N) \sim \left(\tfrac{9}{8}N\right)^{1/2}$.

depends on sorry what this means
Statement uses: Asymptotics.IsEquivalent, Nat, Real, Filter.atTop, Erdos441.g, Real.sqrt, HMul.hMul, HDiv.hDiv, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
34 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_441.variants.chen :
    (fun N : ℕ ↦ (g N : ℝ)) ~[atTop] fun N ↦ √(9 / 8 * N) := by
…
```
Mentions: Erdos441.g
theorem Erdos441.erdos_441.variants.chen_dai_upper_bound Chen and Dai [DaCh06] proved that $g(N)\leq \left(\tfrac{9}{8}N\right)^{1/2}+ O\left(\left(\frac{N}{\log N}\right)^{1/2}\log\log N\right)$ sorry line 129 source
Source documentation
Chen and Dai [DaCh06] proved that $g(N)\leq \left(\tfrac{9}{8}N\right)^{1/2}+ O\left(\left(\frac{N}{\log N}\right)^{1/2}\log\log N\right)$.

depends on sorry what this means
Statement uses: Asymptotics.IsBigO, Nat, Real, Filter.atTop, HSub.hSub, Erdos441.g, Real.sqrt, HMul.hMul, HDiv.hDiv, OfNat.ofNat, Real.log
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
30 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_441.variants.chen_dai_upper_bound :
    (fun N : ℕ ↦ (g N : ℝ) - √(9 / 8 * N)) =O[atTop]
      fun N ↦ √(N / Real.log N) * Real.log (Real.log N) := by
…
```
Mentions: Erdos441.g