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

Jinyan He; Jiagui Luo; Shuanglin Fei; Jinyan He; Jiagui Luo; Shuanglin Fei

doi:10.3934/math.2022401

AIMS Mathematics

2022, Volume 7, Issue 4: 7187-7198. doi: 10.3934/math.2022401

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

1.
School of Mathematics and Information, China West Normal University, Nanchong 637009, China
2.
Mathematical College, Sichuan University, Chengdu 610064, China

Received: 03 November 2021 Revised: 07 January 2022 Accepted: 16 January 2022 Published: 09 February 2022
MSC : 11D61

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.
- exponential Diophantine equations,
- integer solution,
- Fibonacci number
Citation: Jinyan He, Jiagui Luo, Shuanglin Fei. On the exponential Diophantine equation $ (a(a-l)m^{2}+1)^{x}+(alm^{2}-1)^{y} = (am)^{z} $[J]. AIMS Mathematics, 2022, 7(4): 7187-7198. doi: 10.3934/math.2022401

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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.

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