401.lean`

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L4DEL: Lean4 Domain Explanation Language

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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

1 declarations, 1 contain sorry, 0 declared as axioms

Erdős Problem 401

References:

erdosproblems.com/401
[ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

Imports (1)

FormalConjecturesUtil

theorem Erdos401.erdos_401 Is there some function sorry line 50 source
Source documentation
Is there some function $f(r)$ such that $f(r)\to \infty$ as $r\to\infty$, such that, for infinitely many $n$, there exist $a_1,a_2$ with $$a_1+a_2> n+f(r)\log n$$ such that $a_1!a_2! \mid n!2^n3^n\cdots p_r^n$?
It is ambiguous in [ErGr80] what the intended quantifiers are on the variables (they write 'is it true that we can find $a_1+a_2>n+f(r)\log n$...'). Comparing to previous problems such as [728] and [729] it seems most likely that they intended to ask the formulation in the problem statement.
The answer is yes: Barreto and Leeham have used ChatGPT to provide a proof of the stated problem (in fact essentially the same construction as their solution to [729]).

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Real, And, Filter.Tendsto, Filter.atTop, LE.le, OfNat.ofNat, Set.Infinite, Set.ofPred, LT.lt, GT.gt, HAdd.hAdd, HMul.hMul, Real.log, Dvd.dvd, Nat.factorial, HPow.hPow and 4 more
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
38 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem erdos_401 : answer(True) ↔
    ∃ f : ℕ → ℝ, Tendsto f atTop atTop ∧
      ∀ (r : ℕ) (hr : 1 ≤ r),
        {n : ℕ | ∃ a₁ a₂ : ℕ, 0 < a₁ ∧ 0 < a₂ ∧
          (a₁ : ℝ) + a₂ > n + f r * Real.log n ∧
          a₁.factorial * a₂.factorial ∣
            n.factorial * (∏ i ∈ Finset.range r, Nat.nth Nat.Prime i) ^ n}.Infinite := by
…
```