The matrix equation $a{X}^m+b{Y}^n=cI$ over $M_2(\mathbb{Z})$

Abstract

Let $\mathbb{N}$ be the set of all positive integers and let $a,b,c$ be nonzero integers such that $\gcd \left(a,b,c\right)=1$. In this paper, we prove the following three results: (1) the solvability of the matrix equation $a{X}^m+b{Y}^n=cI,X,Y\in M_2(\mathbb{Z}),m,n\in \mathbb{N}$ can be reduced to the solvability of the corresponding Diophantine equation if $XY\ne YX$ and the solvability of the equation $a{x}^m+b{y}^n=c,m,n\in \mathbb{N}$ in quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of the matrix equation $X^n+Y^n=c^{n}I,X,Y\in M_2(\mathbb{Z}),n\in \mathbb{N},n\ge 3$, and the solvability of this matrix equation can be reduced to the solvability of the equation $x^n+y^n=c^n,n\in \mathbb{N},n\ge 3$ in quadratic fields if $XY=YX$; (3) we determine all solutions of the matrix equation $a{X}^2+b{Y}^2=cI,X,Y\in M_2(\mathbb{Z})$.