We obtain necessary and sufficient conditions for solving Diophantine algebraic matrix equations in terms of square circulant matrices. We apply these conditions to show that the Fermat matrix equation Xⁿ+Yⁿ = Zⁿ, n ∈ N, n ≥ 3, has no nontrivial solution of that kind, and to construct solutions of the Markov matrix equation

X² + Y² + Z² = 3XYZ.

Diophantine matrix equations, circulant matrices, Fermat matrix equation, Markov matrix equation.

Received: June 11, 2024; Revised: July 18, 2024; Accepted: July 25, 2024; Published: October 4, 2024

How to cite this article: Claude Gauthier, A note on Diophantine algebraic equations of square circulant matrices, JP Journal of Algebra, Number Theory and Applications 63(6) (2024), 505-516. https://doi.org/10.17654/0972555524030

This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License

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