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 $p\equiv 7(\mathrm{mod}12)$. 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.