Fermat's and Catalan's equations over $M_2(\mathbb{Z})$

Abstract

Let $A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\in M_2\left(\mathbb{Z}\right)$ be a given matrix such that $bc\ne 0$ and let $C(A)=\{B\in M_2(\mathbb{Z}):AB=BA\}$. In this paper, we give a necessary and sufficient condition for the solvability of the matrix equation $u{X}^i+v{Y}^j=w{Z}^k,i,j,k\in \mathbb{N},X,Y,Z\in C(A)$, where $u,v,w$ are given nonzero integers such that $\gcd \left(u,v,w\right)=1$. 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 $M_2\left(\mathbb{Z}\right)$ 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.