PROVED
This has been solved in the affirmative.
Let $t(n)$ be maximal such that there is a representation\[n!=a_1\cdots a_n\]with $t(n)=a_1\leq \cdots \leq a_n$. Obtain good bounds for $t(n)/n$. In particular, is it true that\[\lim \frac{t(n)}{n}=\frac{1}{e}?\]Furthermore, does there exist some constant $c>0$ such that\[\frac{t(n)}{n} \leq \frac{1}{e}-\frac{c}{\log n}\]for infinitely many $n$?
It is easy to see that\[\lim \frac{t(n)}{n}\leq \frac{1}{e}.\]Erdős
[Er96b] wrote he, Selfridge, and Straus had proved a corresponding lower bound, so that $\lim \frac{t(n)}{n}=\frac{1}{e}$, and 'believed that Straus had written up our proof. Unfortunately Straus suddenly died and no trace was ever found of his notes. Furthermore, we never could reconstruct our proof, so our assertion now can be called only a conjecture.'
Alladi and Grinstead
[AlGr77] have obtained similar results when the $a_i$ are restricted to prime powers.
Both questions were answered by Alexeev, Conway, Rosenfeld, Sutherland, Tao, Uhr, and Ventullo
[ACRSTUV25], who proved that\[\frac{t(n)}{n}= \frac{1}{e}-\frac{c_0}{\log n}+O\left(\frac{1}{(\log n)^{1+c}}\right),\]where $c_0=0.3044\cdots$ is an explicit constant, for some $c>0$. They also obtain highly precise computations of $t(n)$ for many $n$, in particular establishing various explicit conjectures of Guy and Selfridge
[GuSe98], such as $t(n)\leq n/e$ for $n\neq 1,2,4$ and $t(n)\geq n/3$ for $n\geq 43632$ (and $43632$ is the best possible here).
This is problem B22 of Guy's collection
[Gu04].
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This page was last edited 28 September 2025.
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #391, https://www.erdosproblems.com/391, accessed 2025-11-16
