Geodesic
On the matrix negative Pell equation
Discussiones Mathematicae. General Algebra and Applications, Tome 29 (2009) no. 1, pp. 35-45.

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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.
Keywords: the matrix negative Pell equation, powers matrices 
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Grytczuk, Aleksander; Kurzydło, Izabela. On the matrix negative Pell equation. Discussiones Mathematicae. General Algebra and Applications, Tome 29 (2009) no. 1, pp. 35-45. http://geodesic.mathdoc.fr/item/DMGAA_2009_29_1_a2/

[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.