On the Solutions of the Diophantine Equation (p+2)^x+4∙p^y=z^2

Abstract

In this paper, we find all non-negative integer solutions (x, y, z) of the Diophantine equation (p+2)^x+4∙p^y=z^2, where p and p+2 are prime numbers with ord_p 2=p-1. We show that if p=3, then the equation has the unique non-negative integer solution, which is (x, y, z)=(1, 0, 3). If p≠3, then the equation has no non-negative integer solution. Moreover, we prove that if p=17, then the equation also has no non-negative integer solution.