On an infinite number of solutions to the Diophantine equation $ x^{n}+y^{p}=z^{q}$ over the square integer matrices

TL;DR

This paper extends the Cayley-Hamilton theorem to identify infinite solutions in square integer matrices for the nonlinear Diophantine equation involving powers of matrices, broadening understanding of matrix solutions to such equations.

Contribution

It introduces a novel extension of the Cayley-Hamilton theorem to find infinite matrix solutions to the Diophantine equation for arbitrary positive exponents.

Findings

Established a family of integer matrices satisfying the equation

Proved the existence of infinitely many solutions

Extended classical theorem to nonlinear matrix equations

Abstract

In this paper, we use some extension of the Cayley-Hamilton theorem to find a family of matrices with integer entries that satisfy the non-linear Diophantine equation $ x^{n}+y^{p}=z^{q}$ where $n,p$ and $q$ are arbitrary positive integers.