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

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

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