On the Diophantine Equation (p + 4n)^ x + p^y = z^2

Abstract

In this paper, we study the Diophantine equation $(p+4n)^x+p^y=z^2,$ where $n$ is a non-negative integer and $p, p+4n$ are prime numbers such that $p\equiv 7\pmod{12}$. We show that the non-negative integer solutions of such equation are $(x, y, z)\in
\{(0, 1, \sqrt {p+1})\} \cup \{ (1, 0, 2\sqrt{n+\frac{p+1}{4}})\}$, where $\sqrt {p+1}$ and $\sqrt{n+\frac{p+1}{4}}$ are integers.