1095.lean`

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

5 declarations, 4 contain sorry, 0 declared as axioms

Erdős Problem 1095

References:

erdosproblems.com/1095
[EES74] Ecklund, Jr., E. F. and Erd\H{o}s, P. and Selfridge, J. L., A new function associated with the prime factors of {$(\sp{n}\sb{k})$}. Math. Comp. (1974), 647--649.
[ELS93] Erdős, P. and Lacampagne, C. B. and Selfridge, J. L., Estimates of the least prime factor of a binomial coefficient. Math. Comp. (1993), 215--224.
[GrRa96] Granville, Andrew and Ramaré, Olivier, Explicit bounds on exponential sums and the scarcity of squarefree binomial coefficients. Mathematika (1996), 73--107.
[Ko99b] Konyagin, S. V., Estimates of the least prime factor of a binomial coefficient. Mathematika (1999), 41--55.
[SSW20] Sorenson, Brianna and Sorenson, Jonathan and Webster, Jonathan, An algorithm and estimates for the {E}rdős-{S}elfridge function. (2020), 371--385.

Imports (1)

FormalConjecturesUtil

def Erdos1095.g Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$ line 48 source
Source documentation
Let $g(k)>k+1$ be the smallest $n$ such that all prime factors of $\binom{n}{k}$ are $>k$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: InfSet.sInf, Set.ofPred, And, LT.lt, HAdd.hAdd, OfNat.ofNat, Nat.minFac, Nat.choose
5 instances, coercions and auxiliary constants left out.
```
noncomputable def g (k : ℕ) : ℕ := sInf {m | k + 1 < m ∧ k < (m.choose k).minFac}
…
```
theorem 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] sorry line 54 source
Source documentation
The current record is $g(k) \gg \exp(c(\log k)^2)$ for some $c>0$, due to Konyagin [Ko99b]. -

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Asymptotics.IsBigO, Nat, Filter.atTop, Real.exp, HMul.hMul, HPow.hPow, Real.log, Erdos1095.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_1095.variants.lower_solved :
    ∃ c > 0, (fun k : ℕ ↦ exp (c * log k ^ 2)) =O[atTop] fun k ↦ (g k : ℝ) := by
…
```
Mentions: c, Erdos1095.g
theorem Erdos1095.erdos_1095.variants.upper_conjecture Ecklund, Erdős, and Selfridge [EES74] proved $g(k)\leq \exp((1+o(1))k)$ sorry line 62 source
Source documentation
Ecklund, Erdős, and Selfridge [EES74] proved $g(k)\leq \exp((1+o(1))k)$.

depends on sorry what this means
Statement uses: Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, nhds, OfNat.ofNat, Filter.Eventually, LE.le, Erdos1095.g, Real.exp, HMul.hMul, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
31 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1095.variants.upper_conjecture :
    ∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧ ∀ᶠ k in atTop, g k ≤ exp (k * (1 + f k)) := by
…
```
Mentions: Erdos1095.g
theorem 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 $c>0$ sorry line 71 source
Source documentation
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 $c>0$.

depends on sorry what this means
Statement uses: Exists, Real, And, GT.gt, OfNat.ofNat, Filter.Eventually, Nat, GE.ge, Erdos1095.g, Real.exp, HDiv.hDiv, HMul.hMul, Real.log, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
25 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1095.variants.lower_conjecture : ∃ c > 0, ∀ᶠ k in atTop, g k ≥ exp (c * k / log k) := by
…
```
Mentions: c, Erdos1095.g
theorem Erdos1095.erdos_1095.variants.log_equivalent Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$ sorry line 78 source
Source documentation
Sorenson, Sorenson, and Webster [SSWE20] give heuristic evidence that $\log g(k) \asymp \frac{k}{\log k}$.

depends on sorry what this means
Statement uses: Asymptotics.IsTheta, Nat, Real, Filter.atTop, Real.log, Erdos1095.g, HDiv.hDiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat
15 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_1095.variants.log_equivalent :
    (fun k ↦ log (g k)) =Θ[atTop] (fun k ↦ (k : ℝ) / log k) := by
…
```
Mentions: Erdos1095.g