On the sparsity of integers $a$ in solutions to $a!b!=c!$

TL;DR

This paper investigates the sparsity of integers involved in solutions to the factorial Diophantine equation a!b!=c!, demonstrating that the set of such a's is sparse and has asymptotic density zero under certain assumptions.

Contribution

It proves that the set of a's appearing in solutions to a!b!=c! is sparse and has density zero, extending previous results on the distribution of c's.

Findings

The set of a's in solutions is sparse.

a cannot be close to a large fraction of primes.

Under equidistribution assumptions, the set of such a's has density zero.

Abstract

We consider the Diophantine equation $$ a!b! = c! $$ due to Erd\H{o}s, where we assume $a \leq b$. It is widely believed that there are only finitely many nontrivial solutions, and considerable work has been dedicated to showing this. In one direction, Luca (2007) showed that the set of $c$'s which can appear in solutions has density zero. Here we show that the set of $a$'s appearing in solutions is also sparse. In particular, $a$ cannot be one less than a large fraction of primes, and, under the assumption that $\sqrt[k]{a!} \mod 1$ is equidistributed in an appropriate sense, we show that the set of such $a$ has asymptotic density zero.