OPEN

Let $k>3$. Does the product of any $k$ consecutive integers $N=\prod_{m< n\leq m+k}n$ (with $m>k$) have a prime factor $p\mid N$ such that $p^2\nmid N$?

Erdős and Selfridge proved that $N$ can never be a perfect power. Erdős remarked that this 'seems hopeless at present'.