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

TL;DR

This work investigates Fermat's and Catalan's Diophantine equations within the 2×2 integer matrix setting by focusing on the centralizer $C(A)$ of a fixed matrix $A$ with $bc\neq 0$. It establishes a bridge between matrix equations in $C(A)$ and scalar Diophantine equations in quadratic fields: a solvability criterion for $uX^i+vY^j=wZ^k$ in $C(A)$ translates to the scalar Fermat equation in $K=\mathbb{Q}(\sqrt{a^2+4bc})$, and solvability of the Catalan matrix equation in $M_2(\mathbb{Z})$ reduces to Catalan’s equation in $\mathcal{O}_K$. The paper analyzes the arithmetic of $C(A)$, showing when it forms an integral domain, and presents corollaries restricting solutions for various $D$; it thus connects matrix Diophantine problems to known number-theoretic problems in quadratic fields. Overall, the results reduce matrix-level questions to classical number theory in quadratic fields, highlighting where nontrivial matrix solutions may exist and where open problems remain.

Abstract

Let $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\in M_2\left(\mathbb{Z}\right)$ be a given matrix such that $bc\neq0$ 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 $uX^i+vY^j=wZ^k,\, i,\, j,\, k\in\mathbb{N},\, X, \,Y,\, Z\in C(A)$, 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.

Theorems & Definitions (35)

• Lemma 2.1
• Lemma 2.2
• Proposition 2.1
• Proposition 2.2
• Corollary 2.1
• Proposition 2.3
• Lemma 3.1
• Lemma 3.2
• Theorem 3.1
• Theorem 3.2