The matrix equation $aX^m+bY^n=cI$ 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 $\gcd(a,\,b,\,c) = 1$. In this paper, we prove the following three results. Firstly, we show that the solvability of the matrix equation $aX^m + bY^n = cI,\,m,\,n \in \mathbb{N}$ in $M_2(\mathbb{Z})$ can be reduced to the solvability of a corresponding Diophantine equation when the matrices $X$ and $Y$ do not commute, i.e., $XY \neq YX$. Alternatively, when $X$ and $Y$ commute, i.e., $XY = YX$, the solvability of this matrix equation can be reduced to the solvability of the equation $ax^m+by^n=c,\, m,\, n\in\mathbb{N}$ in quadratic fields. Secondly, we determine all solutions of the matrix equation $X^n + Y^n = c^nI,\,n \in \mathbb{N},\,n \geq 3$ in $M_2(\mathbb{Z})$ when $X$ and $Y$ do not commute. Moreover, when $X$ and $Y$ commute, we show that the solvability of this matrix equation can be reduced to the solvability of the equation $x^n + y^n = c^n,\,n \in \mathbb{N},\,n \geq 3$ in quadratic fields. Finally, we determine all solutions of the matrix equation $aX^2 + bY^2 = cI$ in $M_2(\mathbb{Z})$.