977.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

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

5 declarations, 5 contain sorry, 0 declared as axioms

Erdős Problem 977

References:

erdosproblems.com/977
[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.
[Sc62] Schinzel, A., On primitive prime factors of $a^n-b^n$. Proc. Cambridge Philos. Soc. (1962), 555-562.
[St74b] Stewart, C. L., The greatest prime factor of $a^n-b^n$. Acta Arith. (1974/75), 427-433.
[St13] Stewart, Cameron L., On divisors of Lucas and Lehmer numbers. Acta Math. (2013), 291-314.
[MuWo02] Murty, Ram and Wong, Siman, The $ABC$ conjecture and prime divisors of the Lucas and Lehmer sequences. (2002), 43-54.
[La21] L. Lai, On the largest prime divisor of $n!+1$. arXiv:2103.14894 (2021).

Imports (1)

FormalConjecturesUtil

theorem Erdos977.erdos_977 sorry line 58 source
depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, HDiv.hDiv, Nat.maxPrimeFac, HSub.hSub, HPow.hPow, OfNat.ofNat, Filter.atTop
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 Erdos977.erdos_977.variants.schinzel Schinzel [Sc62] proved that $P(2^n-1)>2n$ for $n>12$ sorry line 64 source
Source documentation
Schinzel [Sc62] proved that $P(2^n-1)>2n$ for $n>12$.

depends on sorry what this means
Statement uses: Nat, LT.lt, OfNat.ofNat, HMul.hMul, Nat.maxPrimeFac, HSub.hSub, HPow.hPow
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_977.variants.schinzel :
    ∀ n : ℕ, 12 < n → 2 * n < Nat.maxPrimeFac (2 ^ n - 1) := by
…
```
Mentions: Nat.maxPrimeFac
theorem 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 $n$ sorry line 70 source
Source documentation
Stewart [St13] proved that $P(2^n-1)\gg n^{1+\frac{1}{104\log\log n}}$ for all large $n$.

depends on sorry what this means
Statement uses: Exists, Real, And, LT.lt, OfNat.ofNat, Filter.Eventually, Nat, LE.le, HMul.hMul, HPow.hPow, HAdd.hAdd, HDiv.hDiv, Real.log, Nat.maxPrimeFac, HSub.hSub, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
54 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_977.variants.stewart : ∃ c : ℝ, 0 < c ∧ ∀ᶠ n : ℕ in atTop,
    c * (n : ℝ) ^ (1 + 1 / (104 * Real.log (Real.log n))) ≤ Nat.maxPrimeFac (2 ^ n - 1) := by
…
```
Mentions: c, Nat.maxPrimeFac
theorem Erdos977.erdos_977.variants.factorial In [Er65b] Erdős also asks about $P(n!+1)$: is it true that $P(n!+1)/n\to\infty$? sorry line 80 source
Source documentation
In [Er65b] Erdős also asks about $P(n!+1)$: is it true that $P(n!+1)/n\to\infty$? This appears to be open still. Murty and Wong [MuWo02] proved that $P(n!+1)>(1+o(1))n\log n$ assuming the abc conjecture.

depends on sorry what this means
Statement uses: Iff, True, Filter.Tendsto, Nat, Real, HDiv.hDiv, Nat.maxPrimeFac, HAdd.hAdd, Nat.factorial, OfNat.ofNat, Filter.atTop
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
20 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_977.variants.factorial : answer(sorry) ↔
    Tendsto (fun n : ℕ ↦ (Nat.maxPrimeFac (n.factorial + 1) : ℝ) / n) atTop atTop := by
…
```
Mentions: Nat.maxPrimeFac
theorem Erdos977.erdos_977.variants.lai The best-known unconditional result, due to Lai [La21], is that $$\limsup \frac{P(n!+1)}{n} \geq 1+9\log 2\approx 7.238.$$ sorry line 89 source
Source documentation
The best-known unconditional result, due to Lai [La21], is that $$\limsup \frac{P(n!+1)}{n} \geq 1+9\log 2\approx 7.238.$$

depends on sorry what this means
Statement uses: Real, LT.lt, HAdd.hAdd, OfNat.ofNat, HMul.hMul, Nat, Real.log, Filter.Frequently, Nat.maxPrimeFac, Nat.factorial, Filter.atTop
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_977.variants.lai : ∀ C : ℝ, C < 1 + 9 * Real.log 2 →
    ∃ᶠ n : ℕ in atTop, C * n < Nat.maxPrimeFac (n.factorial + 1) := by
…
```
Mentions: Nat.maxPrimeFac