The matrix equation $aX^m+bY^n=cI$ over $M_2(\mathbb{Z})$

TL;DR

This paper investigates solutions to specific matrix equations over 2x2 integer matrices, reducing their solvability to classical Diophantine equations and characterizing solutions in various cases.

Contribution

It provides a comprehensive analysis of the matrix equation $aX^m+bY^n=cI$, linking its solvability to Diophantine equations and classifying solutions for special cases.

Findings

Solvability reduces to Diophantine equations when matrices do not commute.

Explicit solutions are characterized for the equation $X^n+Y^n=c^nI$ in non-commutative cases.

Solutions for $aX^2+bY^2=cI$ are fully determined.

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 $aX^m+bY^n=cI,\,X,\,Y\in M_2(\mathbb{Z}),\, m,\, n\in\mathbb{N}$ can be reduced to the solvability of the corresponding Diophantine equation if $XY\neq YX$ and the solvability of the equation $ax^m+by^n=c,\, m,\, n\in\mathbb{N}$ in quadratic fields if $XY=YX$; (2) we determine all non-commutative solutions of the matrix equation $X^n+Y^n=c^nI,\,X,\,Y\in M_2(\mathbb{Z}),\,n\in\mathbb{N},\,n\geq3$, and the solvability of this matrix equation can be reduced to the solvability of the equation $x^n+y^n=c^n,\, n\in\mathbb{N},\,n\geq3$ in quadratic fields if $XY=YX$; (3) we determine all solutions of the matrix equation $aX^2+bY^2=cI,\,X,\,Y\in M_2(\mathbb{Z})$.