On the Diophantine Equations $q^x + p(2q + 1)^y = z^2$ and $q^x + p(4q + 1)^y = z^2$

Abstract

In this paper, by using basic concepts of number theory, we present some conditions of the non-existence of non-negative integer solutions $(x, y, z)$ for the Diophantine equations $q^x + p(2q + 1)^y = z^2$ and $q^x + p(4q + 1)^y = z^2$, where $p$ and $q$ are prime numbers.