On the Diophantine Equation $p^x + (p+5)^y = z^2$, where $p$ is Odd Prime

Renz Jimwel S. Mina, Jerico B. Bacani

Abstract

This paper studies the Diophantine equation $p^x + (p+5)^y =z^2$, where $p$ is an odd prime number. Results on Legendre and Jacobi symbols are used, and the transformation to an elliptic curve of rank zero is done to show that this equation has a unique positive integer solution $(p,x,y,z) = (5,3,2,15)$ whenever $x \not \equiv 1 \pmod{12}$.