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 $k\ge 2$, odd integer $b$ and an odd prime exponent $q$ and 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 $(b,q)$ when $3\le b<{10}^6$. As an application, we prove that the related equation $(2^k-1)(b^k-1)=x^n,$ has no positive integer solution $(k,x,n)$ for several specific odd values of $b$, including $b\in \{5,7,11,13,21,23,27,29\}$.