Note on the matrix Fermat’s equation

Aleksander Grytczuk and Izabela Kurzydło
Notes on Number Theory and Discrete Mathematics, ISSN 1310-5132
Volume 17, 2011, Number 2, Page 4—11
Download full paper: PDF, 182 Kb

Details

Authors and affiliations

Aleksander Grytczuk
Faculty of Mathematics, Computer Science and Econometrics,
University of Zielona Gora,
65-516 Zielona Gora, Poland

Izabela Kurzydło
Faculty of Mathematics, Computer Science and Econometrics,
University of Zielona Gora,
65-516 Zielona Gora, Poland

Abstract

We consider the Fermat’s equation Xn + Yn = Zn in the set of 2 × 2 rational matrices. We give some necessary condition of solvabillity of this equation.

Keywords

  • The matrix equation
  • The matrix Fermat’s equation
  • Powers of matrices

AMS Classification

  • 15A24
  • 15A42

References

  1. Bialek,K. and Grytczuk, A., 1987, The equation of Fermat in G2(k) and Q(√k), Acta Academiae Paedagogicae Agriensis. Matematika,18/11, 81-90.
  2. Cao, Z. and Grytczuk, A., 1998, Fermat’s type equations in the set of 2×2 integral matrices, Tsukuba J. Math., 22, 637-643.
  3. Cao, Z. and Grytczuk, A., 2000, Some remarks on Fermat’s equation in the set of matrices, Acta Acad. Paed. Agriensis, Sectio Math., 27, 39-46.
  4. Domiaty, R. Z., 1966, Solutions of x4 +y4 = z4 in 2×2 integral matrices, Amer. Math. Monthly, 73, 631.
  5. Grytczuk, A., 1997, Fermat’s equation in the set of matrices and special functions, Studia Univ. Babes-Bolyai, Mathematica, 4, 49-55.
  6. Grytczuk, A., 1998, On a conjecture about the equation Amx + Amy = Amz, Acta Acad. Paed. Agrieusis, Sectio Math., 25, 61-70.
  7. Grytczuk, A., 1995, On Fermat’s equation in the set of integral 2×2 matrices, Period. Math. Hungar., 30, 67-72.
  8. Grytczuk, A. and Kurzydlo, I.,2009, The necessary and sufficient condition for the solvability of Diophantine matrix equation Amx + Amy + Amz = Amw; Boletin de la
    Sociedad Matematica Mexicana.,15, 109-116.
  9. Khazanov, A., 1995, Fermat’s equation in matrices, Serodica Math. J., 21, 19-40.
  10. Le, M. and Li, C., 1995, On Fermat’s equation in integral 2×2 matrices, Period. Math. Hung., 31, 219-222.
  11. Patay, Z. and Szakacs, A., 2002, On Fermat’s problem in matrix rings and groups, Publ. Math. Debrecen, 61/3-4, 487-494.
  12. Qin, H., 1996, Fermat’s problem and Goldbach problem over Mn(Z), Linear Algebra App., 236, 131-135.
  13. Ribenboim, P., 1979, 13 Lectures on Fermat’s Last Theorem (New York: Springer-
    Verlag).
  14. Vaserstein, N., 1989, Non-commutative Number Theory, Contemp. Math., 83, 445-449.

Related papers

Cite this paper

APA

Grytczuk, A., & Kurzydło, I. (2011). Note on the matrix Fermat’s equation. Notes on Number Theory and Discrete Mathematics, 17(2), 4-11.

Chicago

Grytczuk, Aleksander, and Izabela Kurzydło. “Note on the Matrix Fermat’s Equation.” Notes on Number Theory and Discrete Mathematics 17, no. 2 (2011): 4-11.

MLA

Grytczuk, Aleksander, and Izabela Kurzydło. “Note on the Matrix Fermat’s Equation.” Notes on Number Theory and Discrete Mathematics 17.2 (2011): 4-11. Print.

Comments are closed.