Erdős Problem #728: Factorial Divisibility

Erdős problem #728 · erdosproblems.com/728

Whether there are infinitely many integers $a, b, n$ with $a, b \ge \varepsilon n$ such that $a!\cdot b!$ divides $n!\cdot(a+b-n)!$ while $a+b$ exceeds $n$ by more than $C\cdot\log n$.

Result

Proved

Status

Resolved

AI contribution

AI-discovered

Method

Argument

Field

Number Theory

Posed by

Paul Erdős, Ronald Graham, Imre Ruzsa, Ernst Straus

Year posed

1975

Years open

51y

Solved

2026-01-06

Model

Aristotle (Harmonic) + GPT-5.2 Pro

Vendor

Harmonic / OpenAI

Collaborators

Boris Alexeev, Kevin Barreto, Liam Price, Nat Sothanaphan

Verification

Lean-verified

Publication

Announced

Significance

10 / 100*

Disclosed cost

—

Wikipedia

No dedicated article