The board / number theory / #729
Erdős Problem #729
PROVED LEANdatabase updated 2026-01-10
The statement as posed is true, and a proof exists. A solution has also been checked by machine, in Lean.
What the community database records, not our verdict. erdosproblems.com no longer shows statuses (why).
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].
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research solved. The authoritative wording is on erdosproblems.com/729.Prizenone
Topicsnumber theory, factorials
Statement in LeanFormal Conjectures
Solution in Leanyes
Original pageerdosproblems.com/729
Attempts
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