Problem Statement
Let $A(n)$ denote the least value of $t$ such that\[n!=a_1\cdots a_t\]with $a_1\leq \cdots \leq a_t\leq n^2$. Is it true that\[A(n)=\frac{n}{2}-\frac{n}{2\log n}+o\left(\frac{n}{\log n}\right)?\]
Categories:
Number Theory Factorials
Progress
If we change the condition to $a_t\leq n$ it can be shown that\[A(n)=n-\frac{n}{\log n}+o\left(\frac{n}{\log n}\right)\]via a greedy decomposition (use $n$ as often as possible, then $n-1$, and so on). Other questions can be asked for other restrictions on the sizes of the $a_t$.Cambie has observed that a positive answer follows from the result above with $a_t\leq n$, simply by pairing variables together, e.g. taking $a_i'=a_{2i-1}a_{2i}$ (and the lower bound follows from Stirling's approximation).
Source: erdosproblems.com/392 | Last verified: January 14, 2026