OPEN
Is it true that, for every $r$, there is a $k$ such that if $I_1,\ldots,I_r$ are disjoint intervals of consecutive integers, all of length at least $k$, then
\[\prod_{1\leq i\leq r}\prod_{m\in I_i}m\]
is not a perfect power?
Erdős and Selfridge
[ErSe75] proved that the product of consecutive integers is never a power (establishing the case $r=1$). The condition that the intervals be large in terms of $r$ is necessary for $r=2$ - see the constructions in
[363].
See also [363] for the case of squares.
