Problem Statement
Let $f(n)$ be the minimal $m$ such that\[n! = a_1\cdots a_k\]with $n< a_1<\cdots <a_k=m$. Is there (and what is it) a constant $c$ such that\[f(n)-2n \sim c\frac{n}{\log n}?\]
Categories:
Number Theory Factorials
Progress
Erdős, Guy, and Selfridge [EGS82] have shown that\[f(n)-2n \asymp \frac{n}{\log n}.\]Source: erdosproblems.com/390 | Last verified: January 14, 2026