Conjectured by Erdős and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]). Erdős and Selfridge proved that $N$ can never be a perfect power. Erdős remarked that this 'seems hopeless at present'.

In [Er82c] he further conjectures that, if $m,k$ are fixed and $n$ is sufficiently large, then 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.

See also [364].