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

Abstract

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 $b\in\{5,7,11,13,21,23,27,29\}$.