Decomposing a factorial into large factors
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arXiv
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| Autores principales: | , , , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866910100689518592 |
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| author | Alexeev, Boris Conway, Evan Rosenfeld, Matthieu Sutherland, Andrew V. Tao, Terence Uhr, Markus Ventullo, Kevin |
| author_facet | Alexeev, Boris Conway, Evan Rosenfeld, Matthieu Sutherland, Andrew V. Tao, Terence Uhr, Markus Ventullo, Kevin |
| contents | Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erdős and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Decomposing a factorial into large factors Alexeev, Boris Conway, Evan Rosenfeld, Matthieu Sutherland, Andrew V. Tao, Terence Uhr, Markus Ventullo, Kevin Number Theory 11A51 Let $t(N)$ denote the largest number such that $N!$ can be expressed as the product of $N$ integers greater than or equal to $t(N)$. The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erdős, Selfridge, and Straus; but the proof is lost. Here we obtain the more precise asymptotic $$ \frac{t(N)}{N} = \frac{1}{e} - \frac{c_0}{\log N} + O\left( \frac{1}{\log^{1+c} N} \right)$$ for an explicit constant $c_0 = 0.30441901\dots$ and some absolute constant $c>0$, answering a question of Erdős and Graham. For the upper bound, a further lower order term in the asymptotic expansion is also obtained. With numerical assistance, we obtain highly precise computations of $t(N)$ for wide ranges of $N$, establishing several explicit conjectures of Guy and Selfridge on this sequence. For instance, we show that $t(N) \geq N/3$ for $N \geq 43632$, with the threshold shown to be best possible. |
| title | Decomposing a factorial into large factors |
| topic | Number Theory 11A51 |
| url | https://arxiv.org/abs/2503.20170 |