On the Exponential Diophantine Equation $ (M_{pq} )^x+(M_{pq}+1)^y=z^2$

Abstract

In this paper, we consider the number $M_{pq}=p^q-1$, where $p>0$ and $q>1$ are integers, and the Exponential Diophantine equation $(M_{pq} )^x +(M_{pq}+1)^y = z^2$, where $x,\ y$ and $z$ are positive integers. We find the solutions to the title equation expect the case only when both $p$ and $y$ are odd integers.