@article{Feng2022, 
author = {Cheng Feng and Jiagui Luo},
title = {On the exponential Diophantine equation              (                                                  q                              2                l                                      −                          p                              2                k                                              2                n            )        x    +  (      p    k        q    l    n      )    y    =            (                                                  q                              2                l                                      +                          p                              2                k                                              2                n            )        z},
year = {2022},
journal = {AIMS Mathematics},
volume = {7},
number = {5},
pages = {8609-8621},
keywords = {exponential Diophantine equations, positive integer solution, quadratic residue},
url = {https://www.sciopen.com/article/10.3934/math.2022481},
doi = {10.3934/math.2022481},
abstract = {Let    k  ,  l  ,      m    1    ,      m    2   be positive integers and let both    p and    q be odd primes such that        p    k    =      2                  m        1              −      a                  m        2             and        q    l    =      2                  m        1              +      a                  m        2             where    a is odd prime with    a  ≡  5    (  mod    8  ) and    a  ≢  1    (  mod    5  ). In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that the exponential Diophantine equation              (                                                  q                              2                l                                      −                          p                              2                k                                              2                n            )        x    +  (      p    k        q    l    n      )    y    =            (                                                  q                              2                l                                      +                          p                              2                k                                              2                n            )        z   has only the positive integer solution    (  x  ,  y  ,  z  )  =  (  2  ,  2  ,  2  ).}
}