On The Exponential Diophantine Equation $p^{2m} + {(6r+1)}^n = z^{2}$

Abstract

A polynomial equation with two or more unknowns for which the integer solutions are sought out is called a Diophantine equation. When exponents are introduced into the equation, a simple linear Diophantine equation transforms into a more complex exponential Diophantine equation. This paper concentrates on finding the solution to the exponential Diophantine equation $p^{2m} + {(6r+1)}^n = z^{2}$ where $p, m, r, n, z \in \mathbb{Z}^+$. There is no integral solution to the equation when $p$ is an odd prime, $m \leq 5$, $r \leq 25$ and $n \leq 10$.