General Solution of the Diophantine equation involving Mersenne Prime

TL;DR

This paper investigates and provides solutions to a specific exponential Diophantine equation involving Mersenne primes, exploring conditions for solutions and cases with no solutions, contributing to number theory research.

Contribution

The paper offers a general solution framework for the Diophantine equation involving Mersenne primes and identifies cases with no solutions, advancing understanding of such equations.

Findings

Identified conditions under which solutions exist

Presented explicit solutions for certain cases

Proved the absence of solutions in specific scenarios

Abstract

In this article, I study and solve the exponential Diophantine equation $M_p^{x} + (M_q + 1)^{y}= (lz)^2$ where $M_p$ and $M_q$ are Mersenne primes, $l$ is a prime number, and $x,y$, and $z$ are non-negative integers. Several illustrations are presented as well as cases where no solution of the given Diophantine equation is present.