Solutions to Diophantine Equations Involving Primes and Perfect Squares

This paper determines integer solutions ( p, m, z ), where p is a prime number, m is a perfect square, and z is a nonnegative integer, to the Diophantine equations of the form ( p + m ) x - p y = z 2 and p x - ( p + m ) y = z 2 , with x and y not both greater than 1. The analysis relies primarily on classical techniques in number theory, including factoring strategies and modular arithmetic. To structure the investigation, the first equation is thoroughly examined in three distinct cases, that is, when p = 2, when p ≡ 1 (mod 4), and when p ≡ 3 (mod 4). Results are also obtained for the second equation. Our results include complete solution families in many cases and partial findings in others.