Non-existence of Positive Integer Solutions of the Diophantine Equation $p^x+(p+2q)^y=z^2$, where $p$, $q$ and $p+2q$ are Prime Numbers

Abstract

The Diophantine equation $p^x+(p+2q)^y=z^2$, where $p$, $q$ and $p+2q$ are prime numbers, is studied widely. Many authors give $q$ as an explicit prime number and investigate the positive integer solutions and some conditions for non-existence of positive integer solutions. In this work, we gather some conditions for odd prime numbers $p$ and $q$ for showing that the Diophantine equation $p^x+(p+2q)^y=z^2$ has no positive integer solution. Moreover, many examples of Diophantine equations with no positive integer solution are illustrated.