On the Non-Linear Diophantine Equations $4^{x} - a^{y} = dz^{2}$ and $4^{x} + a^{y} = dz^{2}$

Abstract

In this article, we study Diophantine equations $4^{x} - a^{y} = dz^{2}$ and $4^{x} + a^{y} = dz^{2}$ where $a, d, x, y,$ and $z$ are non-negative integers. Under some conditions of integers $a$, $d$ and by using congruence properties, we give all non-negative integer solutions of $4^{x} - a^{y} = dz^{2}$ and we show that $4^{x} + a^{y} = dz^{2}$ has no non-negative integer solution.