On the Diophantine Equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ Regarding Terai's Conjecture

Year 2024, Volume: 7 Issue: 4, 199 - 211, 31.12.2024

Tuba Çokoksen , Murat Alan

https://doi.org/10.33434/cams.1561789

Abstract

This study establishes that the sole positive integer solution to the exponential Diophantine equation $(8r^2+1)^x+(r^2-1)^y=(3r)^z$ is $(x,y,z)=(1,1,2)$ for all $r>1$. The proof employs elementary techniques from number theory, a classification method, and Zsigmondy's Primitive Divisor Theorem.

Keywords

Diophantine equations , Primitive divisor theorem , Terai’s conjecture

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