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arXiv · 2512.03188

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

Joshua Cooper·Joseph Preuss
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Abstract

We consider the Diophantine equation $$ a!b! = c! $$ due to Erdős, 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.

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Joshua Cooper, Joseph Preuss. 2025-12-02. On the sparsity of integers $a$ in solutions to $a!b!=c!$. https://arxiv.org/abs/2512.03188

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