Erdős believed not. Erdős and Selfridge [ErSe75] proved that the product of consecutive integers is never a perfect power.

The theory of Pell equations implies that there are infinitely many pairs $n,d$ with $(n,d)=1$ such that $n(n+d)(n+2d)$ is a square.

Considering the question of whether the product of an arithmetic progression of length $k$ can be equal to an $\ell$th power:

• Euler proved this is impossible when $k=4$ and $\ell=2$,
• Obláth [Ob51] proved this is impossible when $(k,l)=(5,2),(3,3),(3,4),(3,5)$.
• Marszalek [Ma85] proved that this is only possible for $k\ll_d 1$, where $d$ is the common difference of the arithmetic progression.

Jakob Führer has observed this is possible for integers in general, for example $(-6)\cdot(-1)\cdot 4\cdot 9=6^3$.