ON THE DIOPHANTINE EQUATION P x + Q y = Z 2

A Diophantine equation is a polynomial equation involving two or more variables for which integral solutions are sought. An exponential Diophantine equation includes additional variables that appear as exponents. This paper focuses on determining integral solutions to the Diophantine equation px + qy = z2, given that x + y = 5, where p and q are twin primes, cousin primes, sexy primes, or any positive integers. By analyzing the solution patterns for each scenario, we aim to develop theorems and lemmas. The findings in this paper demonstrate that, for all cases where x + y = 5, the Diophantine equation does not have any non-trivial solutions when p and q are twin primes, cousin primes, or sexy primes, but it does have infinitely many solutions for any positive integers.