Skip to main navigation menu Skip to main content Skip to site footer

Vol. 51 No. 2 (2025)
Serdica Mathematical Journal

Articles

A recipe on the construction of matrices satisfying \(X^k+Y^k=Z^k\) over \(M_{n}(\mathbb{Z})\)

https://doi.org/10.55630/serdica.2025.51.189-196

Published 20-10-2025

  • Kuldeep Sarma

Kuldeep Sarma
Tezpur University

Abstract

In this note, we provide a recipe for the construction of matrices satisfying Fermat's matrix equation over \(M_{n}(\mathbb{Z})\).

2020 Mathematics Subject Classification:

15B33, 15A24, 11D41

Keywords

Matrix equation, Fermat equation

Full text

Author Details

Kuldeep Sarma

Department of Applied Sciences
Tezpur University
Napaam, Sonitpur, Assam-784028, India
e-mail: kuldeep.sarma65@gmail.com

References

  1. M. Arnold, A. Eydelzon. On matrix Pythagorean triples. Amer. Math. Monthly 126, 2 (2019), 158–160.
  2. E. D. Bolker. Solutions of Ak + Bk = Ck in n × n integral matrices. Amer. Math. Monthly 75 (1968) 759–760.
  3. J. L. Brenner, B. Jacobson. Problems and Solutions: Solutions of Elementary Problems: E2030. Amer. Math. Monthly 78, 5 (1971), 543–544
  4. M.-T. Chien, J. Meng. Fermat's equation over 2-by-2 matrices, Bull. Korean Math. Soc. 58, 3 (2021) 609–616.
  5. R. Z. Domiaty. Solutions of x4 + y4 = z4 in 2 × 2 integral matrices. Amer. Math. Monthly 73 (1966), 631.
  6. N. Freitas, S. Siksek. Fermat's last theorem over some small real quadratic fields. Algebra Number Theory 9, 4 (2015), 875–895.
  7. P. M. Gibson. Solutions of AK + BK = CK in nonsingular integral matrices. Math. Mag. 43 (1970), 275–276.
  8. A. Grytczuk. On Fermat's equation in the set of integral 2 × 2 matrices. Period. Math. Hungar. 30, 1 (1995) 67–72.
  9. A. Grytczuk, I. Kurzydlo. Note on the matrix Fermat's equation. Notes on Number Theory and Discrete Mathematics 17, 2 (2011) 4–11, https://nntdm.net/papers/nntdm-17/NNTDM-17-2-04-11.pdf.
  10. A. Khazanov. Fermat's equation in matrices. Serdica Math. J. 21, 1 (1995) 19–40.
  11. M. Le, C. Li. On Fermat's equation in integral 2 × 2 matrices. Period. Math. Hungar. 31, 3 (1995) 219–222.
  12. L. J. Mordell. Diophantine equations. Pure Appl. Math., vol. 30. London-New York, Academic Press, 1969.
  13. H. Qin. Fermat's problem and Goldbach's problem over Mn(Z). Linear Algebra Appl. 236 (1996), 131–135.
  14. L. N. Vaserstein. Noncommutative number theory. Contemp. Math. 83 (1989) 445–449.
  15. A. Wiles. Modular elliptic curves and Fermat's last theorem. Ann. of Math. (2) 141, 3 (1995), 443–551.