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

Authors : Murat ALAN, Ruhsar Gizem BİRATLI

Pages : 174-180

Doi:10.33401/fujma.1038699

View : 20 | Download : 14

Publication Date : 2022-09-23

Article Type : Research Paper

Abstract :Let $m$ be a positive integer. In this paper, we consider the exponential Diophantine equation $insert ignore into journalissuearticles values(6m^{2}+1);^{x}+insert ignore into journalissuearticles values(3m^{2}-1);^{y}=insert ignore into journalissuearticles values(3m);^{z}$ and we show that it has only unique positive integer solution $insert ignore into journalissuearticles values(x,y,z);=insert ignore into journalissuearticles values(1,1,2);$ for all $ m>1. $ The proof depends on some results on Diophantine equations and the famous primitive divisor theorem.
Keywords : Classification method, Exponential Diophantine equations, Primitive divisor theorem, Terais conjecture