FormalConjectures.ErdosProblems.«728»

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import FormalConjecturesUtil

Erdős Problem 728

Reference: erdosproblems.com/728


open Real

open scoped Nat Topology


namespace Erdos728

Let $\varepsilon$ be sufficiently small and $C, C' > 0$. Are there integers $a, b, n$ such that $$a, b > \varepsilon n\quad a!, b! \mid n!, (a + b - n)!, $$ and $$C \log n < a + b - n < C' \log n ?$$

Note that the website currently displays a simpler (trivial) version of this problem because $a + b$ isn't assumed to be in the $n + O(\log n)$ regime.

Barreto and ChatGPT-5.2 have proved that, for any $0 < C_1 < C_2$, there are infinitely many $a, b, n$ with $b = n/2$, $a = n/2 + O(\log n)$, and $C_1 \log n < a + b - n < C_2 \log n$ such that $a! b! \mid n! (a + b - n)!$

This appears to answer the question in the spirit it was intended.

This was formalized in Lean by Alexeev using Aristotle.

@[category research solved, AMS 11, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.24.0/ErdosProblems/Erdos728p.lean"]
theorem declaration uses 'sorry'erdos_728 :
    answer(True) ↔
      ∀ᶠ ε : ℝ in 𝓝[>] 0, ∀ C > (0 : ℝ), ∀ C' > C,
        ∃ a b n : ℕ,
          0 < n ∧
          ε * n < a ∧
          ε * n < b ∧
          a ! * b ! ∣ n ! * (a + b - n)! ∧
          a + b > n + C * log n ∧
          a + b < n + C' * log n := by⊢ True ↔
  ∀ᶠ (ε : ℝ) in 𝓝[>] 0,
    ∀ C > 0,
      ∀ C' > C,
        ∃ a b n,
          0 < n ∧
            ε * ↑n < ↑a ∧
              ε * ↑n < ↑b ∧ a ! * b ! ∣ n ! * (a + b - n)! ∧ ↑a + ↑b > ↑n + C * log ↑n ∧ ↑a + ↑b < ↑n + C' * log ↑n
  sorryAll goals completed! 🐙


-- TODO(firsching): Use Legendre's formula to test divisibility in terms of p-adic valuations.

end Erdos728