@article{Laipaporn2025, 
author = {Kittipong Laipaporn and Saeree Wananiyakul and Prathomjit Khachorncharoenkul},
title = {Explicit solutions and non-solutions for the Diophantine equation        p    x    +      q          2      y        =      z          2      n       involving primes    p  ≢  q    (  mod    4  )},
year = {2025},
journal = {AIMS Mathematics},
volume = {10},
number = {7},
pages = {15720-15736},
keywords = {Diophantine equation, Catalan's conjecture, Legendre symbol},
url = {https://www.sciopen.com/article/10.3934/math.2025704},
doi = {10.3934/math.2025704},
abstract = {Over the past decade, significant research has been conducted on the equation        a    x    +      b    y    =      z    2   under various conditions imposed on    a and    b or on    x and    y. Most studies focus on conditions where the equation has no solution, while some explore cases with infinitely many solutions, often considering scenarios where    x or    y is even. Motivated by this line of inquiry, we have been inspired to investigate and analyze equations of the form        p    x    +      q          2      y        =      z          2      n       for two distinct primes    p and    q, and to present explicit forms of their solutions    (  p  ,  x  ,  q  ,  y  ,  z  ,  n  ). Recent studies on the exponential Diophantine equation        p    x    +      q    y    =      z    2  , where    p and    q are primes, have addressed cases where    p  =  2 or    p  ≡  q    (  mod    4  ). In this paper, we address the case where    p  ≢  q    (  mod    4  ) and    y is even. In addition, we explore special cases where    z is the prime and provide the complete set of solutions for        p    x    +      q          2      y        =      z          2      n      . We also show that the equation has no solution when    {  2  ,  3  }  ⊈  {  p  ,  q  ,  z  }. In other words, we provide almost explicit solutions to        p    x    +      q          y        =      z          2      n       except for the case where both    x and    y are odd.}
}