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On the matrix negative Pell equation

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EN
Abstrakty
EN
Let N be a set of natural numbers and Z be a set of integers. Let M₂(Z) denotes the set of all 2x2 matrices with integer entries.
We give necessary and suficient conditions for solvability of the matrix negative Pell equation
(P) X² - dY² = -I with d ∈ N
for nonsingular X,Y belonging to M₂(Z) and his generalization
(Pn) $∑_{i=1}^{n} X₂_{i} - d ∑_{i=1}^{n} Y²_{i} = -I$ with d ∈ N
for nonsingular $X_{i},Y_{i} ∈ M₂(Z)$, i=1,...,n.
Twórcy
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
  • Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland
Bibliografia
  • [1] Z. Cao and A. Grytczuk, Fermat's type equations in the set of 2x2 integral matrices, Tsukuba J. Math. 22 (1998), 637-643.
  • [2] R.Z. Domiaty, Solutions of x⁴+y⁴=z⁴ in 2x2 integral matrices, Amer. Math. Monthly (1966) 73, 631.
  • [3] A. Grytczuk, Fermat's equation in the set of matrices and special functions, Studia Univ. Babes-Bolyai, Mathematica 4 (1997), 49-55 .
  • [4] A. Grytczuk, On a conjecture about the equation $A^{mx} + A^{my} =A^{mz}$, Acta Acad. Paed. Agriensis, Sectio Math. 25 (1998), 61-70.
  • [5] A. Grytczuk and J. Grytczuk, Ljunggren's trinomials and matrix equation $A^{x} + A^{y} = A^{z}$, Tsukuba J. Math. 2 (2002), 229-235.
  • [6] A. Grytczuk and K. Grytczuk, Functional recurences, 115-121 in: Applications of Fibonacci Numbers, Ed. E. Bergum et als, by Kluwer Academic Publishers 1990.
  • [7] A. Grytczuk, F. Luca and M. Wójtowicz, The negative Pell equation and Pythagorean triples, Proc. Japan Acad. 76 (2000), 91-94.
  • [8] A. Khazanov, Fermat's equation in matrices, Serdica Math. J. 21 (1995), 19-40.
  • [9] I. Kurzydło, Explicit form on a GLW criterion for solvability of the negative Pell equation - Submitted.
  • [10] M. Le and C. Li, On Fermat's equation in integral 2x2 matrices, Period. Math. Hung. 31 (1995), 219-222.
  • [11] Z. Patay and A. Szakacs, On Fermat's problem in matrix rings and groups, Publ. Math. Debrecen 61 (3-4) (2002), 487-494.
  • [12] H. Qin, Fermat's problem and Goldbach problem over $M_{n}(Z)$, Linear Algebra App., 236 (1996), 131-135.
  • [13] P. Ribenboim, 13 Lectures on Fermat's Last Theorem (New York: Springer-Verlag) 1979.
  • [14] N. Vaserstein, Non-commutative Number Theory, Contemp. Math. 83 (1989), 445-449.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1150
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