For every real constant C>0, does there exist an integer K≥3 such that there are infinitely many triples (a,b,n)∈ N^3 satisfying a>0, b>0, n>0, a+b>n+Clog n, and the denominator of the rational number fracn!a!b! has no prime divisor greater than K? Equivalently, for every prime p>K, the p-adic valuation of that denominator is zero.

Erdős Problem #729 — Large factorial ratios with bounded large prime factors

The problem asks whether, for every constant $C>0$ , infinitely many triples satisfy $a+b>n+C\log n$ while the denominator of $n!/(a!b!)$ has only primes bounded in terms of $C$ . Erdős proved in 1968 that $a!b!\mid n!$ forces $a+b\le n+O(\log n)$ ; the problem asks whether this bound can be exceeded by every fixed logarithmic amount.

Known results

Erdős, 1968: $a!b!\mid n!$ implies $a+b\le n+O(\log n)$ .
Carl Pomerance subsequently noted that a related logarithmic-gap result follows by modifying his earlier argument, with a note covering $k\le .7\log m$ .

January 2026 claimed solution

Barreto and Leeham announced an affirmative solution, obtained from an argument for Problem #728 and subsequently autoformalized by Aristotle from a GPT-5.2 Pro proof. The associated writeup states the required logarithmic-gap factorial-divisibility result, but the available repository extract still displays by sorry; independent mathematical verification is therefore lacking.

Current status (as of June 2026): A positive AI-generated solution and claimed formalization exist, but the problem remains unverified pending an independently checkable completed proof.

Erdős Problem #729 — Large factorial ratios with bounded large prime factors

Primary source

Additional references

Progress summary

Solutions 0