Erdős Problem #396

The problem

Is it true that for every $k$ there exists $n$ such that

$$\prod_{0\leq i\leq k}(n-i) \mid \binom{2n}{n}?$$

Statement quoted from erdosproblems.com/396, maintained by Thomas Bloom. Original source: [ErGr80]. Current status there: open.

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.

∎