On the sparsity of integers $a$ in solutions to $a!b!=c!$
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918229618720768 |
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| author | Cooper, Joshua Preuss, Joseph |
| author_facet | Cooper, Joshua Preuss, Joseph |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03188 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the sparsity of integers $a$ in solutions to $a!b!=c!$ Cooper, Joshua Preuss, Joseph Number Theory 11B65 (Primary) 11A07, 11K36 (Secondary) 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. |
| title | On the sparsity of integers $a$ in solutions to $a!b!=c!$ |
| topic | Number Theory 11B65 (Primary) 11A07, 11K36 (Secondary) |
| url | https://arxiv.org/abs/2512.03188 |