Let $k\geq 3$. Can the product of any $k$ consecutive integers $N$ ever be powerful? That is, must there always exist a prime $p\mid N$ such that $p^2\nmid N$?

Erdős [Er82c] conjectures that, if $k$ is fixed, then for all $n$ sufficiently large and all positive integers $m$, there must be at least $k$ distinct primes $p$ such that $p\mid m(m+1)\cdots (m+n)$ and yet $p^2$ does not divide the right hand side.

[Er82c] Erdős, Paul, "Miscellaneous problems in number theory". Congr. Numer. (1982), 25-45.,