The Diophantine equation $(2^{k}-1)(b^{k}-1)=y^{q}$

In this paper, we consider the exponential Diophantine equation $(2^k-1)(b^k-1)=y^q$, with $k\ge 2$, an odd integer $b$, and an odd prime exponent $q$. We obtain effective upper bounds for $q$ in terms of $b$. In particular, we show that $q\le\log_2(b+1)$ holds apart from a finite, explicitly determined set of exceptional pairs $(b,q)$ with $3\le b<10^6$. As an application, we prove that the related equation $(2^k-1)(b^k-1)=x^n$ has no positive integer solution $(k,x,n)$ with $k,n\geq2$ for several specific values of $b$, including $b\in\{5,7,11,13,21,23,27,29\}$, except for $(2^2-1)(7^2-1)=12^2$.