On the matrix negative Pell equation

• Autorzy: Aleksander Grytczuk , Izabela Kurzydło
• Języki publikacji: 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.

Słowa kluczowe

EN

the matrix negative Pell equation; powers matrices

Kategorie tematyczne

• 15A42: Inequalities involving eigenvalues and eigenvectors
• 15A24: Matrix equations and identities

• Wydawca: Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra
• Czasopismo: Discussiones Mathematicae - General Algebra and Applications
• Rocznik: 2009
• Tom: 29
• Numer: 1
• Strony: 35-45

Daty

wydano

2009

otrzymano

2009-02-05

poprawiono

2009-03-10

Twórcy

autor

Aleksander Grytczuk

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland

autor

Izabela Kurzydło

• Faculty of Mathematics, Computer Science and Econometrics, University of Zielona Góra, prof. Z. Szafrana 4a, 65-516 Zielona Góra, Poland

Bibliografia

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• [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.
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• [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.
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• [12] H. Qin, Fermat's problem and Goldbach problem over $M_{n}(Z)$, Linear Algebra App., 236 (1996), 131-135.
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Identyfikatory

DOI

10.7151/dmgaa.1150