Erdős Problem #686

The problem

Can every integer $N\geq 2$ be written as

$$N=\frac{\prod_{1\leq i\leq k}(m+i)}{\prod_{1\leq i\leq k}(n+i)}$$

for some $k\geq 2$ and $m\geq n+k$?

Statement quoted from erdosproblems.com/686, maintained by Thomas Bloom. Original source: [Er79d]. 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.

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