On the Exponential Diophantine Equation $(6m^{2}+1)^{x}+(3m^{2}-1)^{y}=(3m)^{z}$

Abstract

Let $m$ be a positive integer. In this paper, we consider the exponential Diophantine equation $(6m^{2}+1)^{x}+(3m^{2}-1)^{y}=(3m)^{z}$ and we show that it has only unique positive integer solution $(x,y,z)=(1,1,2)$ for all $ m>1. $ The proof depends on some results on Diophantine equations and the famous primitive divisor theorem.