Erdős Problem 729

References:

• erdosproblems.com/729
• [EGRS75] Erdős, P. and Graham, R. L. and Ruzsa, I. Z. and Straus, E. G., On the prime factors of $\binom{2n}{n}$. Math. Comp. (1975), 83-92.
• [Er68c] Erdős, P., Aufgabe 557. Elemente Math. (1968), 111-113.

theorem Erdos729.erdos_729 : True ↔ ∀ C > 0, ∃ K ≥ 3, {(a, b, n) : ℕ × ℕ × ℕ | a > 0 ∧ b > 0 ∧ n > 0 ∧ ↑a + ↑b > ↑n + C * Real.log ↑n ∧ ∀ (p : ℕ), Nat.Prime p → p > K → padicValNat p (↑n.factorial / (↑a.factorial * ↑b.factorial)).den = 0}.Infinite

Let $C>0$ be a constant. Are there infinitely many integers $a,b,n$ with $a+b> n+C\log n$ such that the denominator of[\frac{n!}{a!b!}]contains only primes $\ll_C 1$?

Erdős [Er68c] proved that if $a!b!\mid n!$ then $a+b\leq n+O(\log n)$. This has been proved in the affirmative by Barreto and Leeham, using ChatGPT and Aristotle, with a modification of the argument used for [728].