On the Diophantine equation $\prod_{i=1}^{m}(a_{i}^{k}-1)=y^{n}$

In this paper, we consider the Diophantine equation $\prod_{i=1}^{m}(a_{i}^{k}-1)=y^{n}$, where $m\geq3$, $a_{1},a_{2},\dots,a_{m}$ are fixed integers which are pairwise coprime. We have proved that the equation has only finitely many solutions $(k,y,n)$ with $n>1$. We also study some special cases of equation when $m=3$ and $m=4$.