On the Diophantine equation $4(7^x)-p^y=z^2$

Abstract

This paper determines all non-negative integer solutions to the Diophantine equation $4(7^x)-p^y=z^2$ where $p$ is a prime. Using modular arithmetic and congruence arguments, we classify all solutions as follows: a unique solution for p = 2, an infinite family of solutions for p = 3, no solutions for 5 ≤ p ≤ 17, and–for p ≥ 19–the existence of solutions requires that $p\not\equiv 19 \pmod{24}$ subject to specific modular constraints. Computational results support the conjecture that no further solutions exist beyond those identified. This work illustrates how modular techniques can fully resolve an exponential Diophantine equation and offers a framework for analyzing similar equations involving mixed exponential and polynomial terms.