Erdős Problem #390

The problem

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

$$f(n)-2n \sim c\frac{n}{\log n}?$$

Statement quoted from erdosproblems.com/390, maintained by Thomas Bloom. Original source: [ErGr80]. Current status there: open (Lean).

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

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