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arXiv · 2503.20170

Decomposing a factorial into large factors

Boris Alexeev·Evan Conway·Matthieu Rosenfeld·Andrew V. Sutherland·Terence Tao·Markus Uhr·Kevin Ventullo
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Abstract

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 The bound $t(N)/N = 1/e-o(1)$ was apparently established in unpublished work of Erd\H{o}s, Selfridge, and Straus; but the proof is lost.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 Erd\H{o}s 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.

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Boris Alexeev, Evan Conway, Matthieu Rosenfeld, Andrew V. Sutherland, Terence Tao, Markus Uhr, Kevin Ventullo. 2025-03-26. Decomposing a factorial into large factors. https://arxiv.org/abs/2503.20170

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