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

ALAN, Murat; BİRATLI, RUHSAR

doi:10.33401/fujma.1038699

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

ALAN M., BİRATLI R. G.

Fundamental journal of mathematics and applications (Online), vol.5, no.3, pp.174-180, 2022 (TRDizin)

Publication Type: Article / Article
Volume: 5 Issue: 3
Publication Date: 2022
Doi Number: 10.33401/fujma.1038699
Journal Name: Fundamental journal of mathematics and applications (Online)
Journal Indexes: TR DİZİN (ULAKBİM)
Page Numbers: pp.174-180
Open Archive Collection: AVESIS Open Access Collection
Yıldız Technical University Affiliated: Yes

Abstract

Let m" role="presentation" >m $m$ be a positive integer. In this paper, we consider the exponential Diophantine equation (6m2+1)x+(3m2−1)y=(3m)z" role="presentation" >(6m2+1)x+(3m2−1)y=(3m)z $(6 m^{2} + 1)^{x} + (3 m^{2} - 1)^{y} = (3 m)^{z}$ and we show that it has only unique positive integer solution (x,y,z)=(1,1,2)" role="presentation" >(x,y,z)=(1,1,2) $(x, y, z) = (1, 1, 2)$ for all m>1." role="presentation" >m>1. $m > 1.$ The proof depends on some results on Diophantine equations and the famous primitive divisor theorem.