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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2512.03188 |
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Table of Contents:
- 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.