Problem 391

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 $n$ ?

Source: erdosproblems.com/391

Number theory
Factorials

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SolvedProved

An accepted solution exists. The statement is true.

The site labels the problem PROVED (LEAN), crediting Alexeev, Conway, Rosenfeld, Sutherland, Tao, Uhr and Ventullo with answering both questions. The standing derived from the claim pages is solved, proved, by the accepted claim Alexeev and others 2025; the Lean behind the site's qualification is third-party work not built here.

ACRSTUV25: B. Alexeev, E. Conway, M. Rosenfeld, A. Sutherland, T. Tao, M. Uhr, and K. Ventullo, Decomposing a factorial into large factors. arXiv:2503.20170 (2025); Math. Comp., in press (2026), DOI 10.1090/mcom/4249.
AlGr77: Alladi, Krishnaswami and Grinstead, Charles, On the decomposition of $n!$ into prime powers. J. Number Theory (1977), 452-458.
Er96b: Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.
Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp. Section B22 "Factorial $n$ as the product of $n$ large factors", printed p. 122: the problem of Straus, Erdős and Selfridge, the example $n=56$ , $l=15$ , Selfridge's two conjectures, and Straus's reputed $l>n/(e+\epsilon)$ whose proof was not found in his Nachlaß. Library home: Guy 2004.
GuSe98: Guy, Richard K. and Selfridge, John L., Unsolved Problems: Factoring Factorial n. Amer. Math. Monthly (1998), 766-767.

Problem 391

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