On the exponential Diophantine equation $ (a(a-l)m^{2}+1)^{x}+(alm^{2}-1)^{y} = (am)^{z} $

Abstract

Suppose that $ a $, $ l $, $ m $ are positive integers with $ a\equiv1\pmod2 $ and $ a^{2}m^{2}\equiv-2\pmod p $, where $ p $ is a prime factor of $ l $. In this paper, we prove that the title exponential Diophantine equation has only the positive integer solution $ (x, y, z) = (1, 1, 2) $. As an another result, we show that if $ a = l $, then the title equation has positive integer solutions $ (x, y, z) = (n, 1, 2) $, $ n\in\mathbb{N} $. The proof is based on elementary methods, Bilu-Hanrot-Voutier Theorem on primitive divisors of Lehmer numbers, and some results on generalized Ramanujan-Nagell equations.

Keywords

• exponential diophantine equations
• integer solution
• fibonacci number