@article{Fei2024, 
author = {Shuanglin Fei and Guangyan Zhu and Rongjun Wu},
title = {On a conjecture concerning the exponential Diophantine equation    (  a      n          2        +  1      )          x        +  (  b      n          2        −  1      )          y        =  (  c  n      )          z},
year = {2024},
journal = {Electronic Research Archive},
volume = {32},
number = {6},
pages = {4096-4107},
keywords = {linear forms in m-adic logarithms, exponential Diophantine equation, positive integer solution},
url = {https://www.sciopen.com/article/10.3934/era.2024184},
doi = {10.3934/era.2024184},
abstract = {Let    a,    b,    c, and    n be positive integers such that    a  +  b  =      c          2      ,    2  ∤  c and    n  >  1. In this paper, we prove that if    gcd  (  c  ,  n  )  =  1 and    n  ≥  117.14  c, then the equation    (  a      n          2        +  1      )          x        +  (  b      n          2        −  1      )          y        =  (  c  n      )          z       has only the positive integer solution    (  x  ,  y  ,  z  )  =  (  1  ,  1  ,  2  ) under the assumption    gcd  (  a      n          2        +  1  ,  b      n          2        −  1  )  =  1. Thus, we affirm that the conjecture proposed by Fujita and Le is true in this case. Moreover, combining the above result with some existing results and a computer search, we show that, for any positive integer    n, if    (  a  ,  b  ,  c  )  =  (  12  ,  13  ,  5  ),    (  18  ,  7  ,  5  ), or    (  44  ,  5  ,  7  ), then this equation has only the solution    (  x  ,  y  ,  z  )  =  (  1  ,  1  ,  2  ). This result extends the theorem of Terai and Hibino gotten in 2015, that of Alan obtained in 2018, and Hasanalizade's theorem attained recently.}
}