Erdős 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 \[\lim \frac{t(n)}{n}=\frac{1}{e}?\] 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 \(c>0\).