On the Diophantine Equation $\\left(9d^2 + 1\\right)^x + \\left(16d^2 - 1\\right)^y = (5d)^z$ Regarding Terai\'s Conjecture

Authors : Tuba Çokoksen, Murat Alan

Pages : 72-84

Doi:10.53570/jnt.1479551

View : 146 | Download : 122

Publication Date : 2024-06-30

Article Type : Research

Abstract :This study proves that the Diophantine equation $\\left(9d^2+1\\right)^x+\\left(16d^2-1\\right)^y=(5d)^z$ has a unique positive integer solution $(x,y,z)=(1,1,2)$, for all $d>1$. The proof employs elementary number theory techniques, including linear forms in two logarithms and Zsigmondy\'s Primitive Divisor Theorem, specifically when $d$ is not divisible by $5$. In cases where $d$ is divisible by $5$, an alternative method utilizing linear forms in p-adic logarithms is applied.
Keywords : Terai, Diophantine equations, primitive divisor theorem