Problem #137: Let $k\geq 3$. Can the product of any $k$ consecutive...

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Progress

Conjectured by Erdős and Selfridge. There are infinitely many $n$ such that $n(n+1)$ is powerful (see [364]). Erdős and Selfridge [ErSe75] proved that $N$ can never be a perfect power. Erdős remarked that this 'seems hopeless at present'.

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

See also [364].