@article{He2022, 
author = {Jinyan He and Jiagui Luo and Shuanglin Fei},
title = {On the exponential Diophantine equation    (  a  (  a  −  l  )      m          2        +  1      )          x        +  (  a  l      m          2        −  1      )          y        =  (  a  m      )          z},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {4},
pages = {7187-7198},
keywords = {exponential Diophantine equations, integer solution, Fibonacci number},
url = {https://www.sciopen.com/article/10.3934/math.2022401},
doi = {10.3934/math.2022401},
abstract = {Suppose that    a,    l,    m are positive integers with    a  ≡  1    (  mod    2  ) and        a          2            m          2        ≡  −  2    (  mod    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  ∈      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.}
}