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A NOTE ON DIOPHANTINE ALGEBRAIC EQUATIONS OF SQUARE CIRCULANT MATRICES
Keywords:
Diophantine matrix equations, circulant matrices, Fermat matrix equation, Markov matrix equationDOI:
https://doi.org/10.17654/0972555524030Abstract
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 $X^n+Y^n=Z^n, n \in \mathbf{N}, n \geq 3$, has no nontrivial solution of that kind, and to construct solutions of the Markov matrix equation
$$
X^2+Y^2+Z^2=3 X Y Z
$$
Received: June 11, 2024
Revised: July 18, 2024
Accepted: July 25, 2024
References
C. Gauthier and G. Kientega, Solutions of some nonlinear Diophantine matrix equations, JP Journal of Algebra, Number Theory and Applications 14(2) (2009), 157-175.
P. J. Davis, Circulant Matrices, AMS Chelsea, Providence, RI, 2013.
I. Kra and S. R. Simarca, On circulant matrices, Notices AMS 59 (2012), 368-377.
I. Stewart and D. Tall, Algebraic Number Theory and Fermat’s Last Theorem, 3rd ed., A K Peters, Natick, MA, 2002.
J. M. Mouanda, J. R. Tsiba and K. Kangni, On Fermat’s last theorem matrix version and galaxies of sequences of circulant matrices with positive integers as entries, Global Journal of Science Frontier Research 22 (2022), 37-65.
A. Markoff, Sur les formes quadratiques binaires indéfinies, Mathematische Annalen 15 (1879), 381-407.
A. Markoff, Sur les formes quadratiques binaires indéfinies (Second mémoire), Mathematische Annalen 17 (1880), 379-399.
M. Aigner, Markov’s Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2015.
M. van Son, Extended Markov number and integer geometry, Ph. D. thesis, University of Liverpool, 2020.
Y. Gyoda and K. Matsushita, Generalization of Markov Diophantine equation via generalized cluster algebra, Electron. J. Combin. 30 (2023), Paper No. 4.10, 20 pp.
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