@article{Li2023, 
author = {Hongjian Li and Pingzhi Yuan},
title = {Fermat's and Catalan's equations over        M    2    (      Z    )},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {1},
pages = {977-996},
keywords = {matrix equation, Fermat's matrix equation, Catalan's matrix equation, quadratic fields},
url = {https://www.sciopen.com/article/10.3934/math.2023047},
doi = {10.3934/math.2023047},
abstract = {Let    A  =      (                            a                          b                                      c                          d                      )    ∈      M    2        (          Z        )   be a given matrix such that    b  c  ≠  0 and let    C  (  A  )  =  {  B  ∈      M    2    (      Z    )  :  A  B  =  B  A  }. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation    u      X    i    +  v      Y    j    =  w      Z    k    ,    i  ,    j  ,    k  ∈      N    ,    X  ,    Y  ,    Z  ∈  C  (  A  ), where    u  ,    v  ,    w are given nonzero integers such that    gcd      (          u      ,            v      ,            w        )    =  1. From this, we get a necessary and sufficient condition for the solvability of the Fermat's matrix equation in    C  (  A  ). Moreover, we show that the solvability of the Catalan's matrix equation in        M    2        (          Z        )   can be reduced to the solvability of the Catalan's matrix equation in    C  (  A  ), and finally to the solvability of the Catalan's equation in quadratic fields.}
}