@article{Fan2023, 
author = {Nan Fan and Jiagui Luo},
title = {On the conjecture of Je                     s            ´      manowicz},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {6},
pages = {14232-14252},
keywords = {exponential Diophantine equations, quadratic residue, positive integer solution},
url = {https://www.sciopen.com/article/10.3934/math.2023728},
doi = {10.3934/math.2023728},
abstract = {Let    k  ,  l  ,      m    1   and        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 a positive integer with    a  ≡      ±    3      (  mod    8  ). In this paper, using only the elementary methods of factorization, congruence methods and the quadratic reciprocity law, we show that Je             s      ´      manowicz' a conjecture holds for the following set of primitive Pythagorean numbers:                       q                  2          l                    −              p                  2          k                      2    ,      p    k        q    l    ,                    q                  2          l                    +              p                  2          k                      2    .We also prove that Je             s      ´      manowicz' conjecture holds for non-primitive Pythagorean numbers:     n                    q                  2          l                    −              p                  2          k                      2    ,  n      p    k        q    l    ,  n                    q                  2          l                    +              p                  2          k                      2    ,for any positive integer    n if for    a  =      a    1        a    2   with        a    1    ≡  1    (  mod    8  ) not a square and    gcd  (      a    1    ,      a    2    )  =  1, then there exists a prime divisor    P of        a    2   such that        (                  a        1            P        )    =  −  1 and    2      |        m    1    ,  a  ≡  5    (  mod    8  ) or    2            ⧸            |        m    2    ,  a  ≡  3    (  mod    8  ).}
}