# On the matrix negative Pell equation

• Volume: 29, Issue: 1, page 35-45
• ISSN: 1509-9415

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## Abstract

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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. 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) ${\sum }_{i=1}^{n}X{₂}_{i}-d{\sum }_{i=1}^{n}Y{²}_{i}=-I$ with d ∈ N for nonsingular ${X}_{i},{Y}_{i}\in M₂\left(Z\right)$, i=1,...,n.

## How to cite

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Aleksander Grytczuk, and Izabela Kurzydło. "On the matrix negative Pell equation." Discussiones Mathematicae - General Algebra and Applications 29.1 (2009): 35-45. <http://eudml.org/doc/276938>.

@article{AleksanderGrytczuk2009,
abstract = {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.},
author = {Aleksander Grytczuk, Izabela Kurzydło},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {the matrix negative Pell equation; powers matrices; matrix negative Pell equation; solvability conditions},
language = {eng},
number = {1},
pages = {35-45},
title = {On the matrix negative Pell equation},
url = {http://eudml.org/doc/276938},
volume = {29},
year = {2009},
}

TY - JOUR
AU - Aleksander Grytczuk
AU - Izabela Kurzydło
TI - On the matrix negative Pell equation
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2009
VL - 29
IS - 1
SP - 35
EP - 45
AB - 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.
LA - eng
KW - the matrix negative Pell equation; powers matrices; matrix negative Pell equation; solvability conditions
UR - http://eudml.org/doc/276938
ER -

## References

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6. [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. Zbl0725.34002
7. [7] A. Grytczuk, F. Luca and M. Wójtowicz, The negative Pell equation and Pythagorean triples, Proc. Japan Acad. 76 (2000), 91-94. Zbl0971.11013
8. [8] A. Khazanov, Fermat's equation in matrices, Serdica Math. J. 21 (1995), 19-40.
9. [9] I. Kurzydło, Explicit form on a GLW criterion for solvability of the negative Pell equation - Submitted.
10. [10] M. Le and C. Li, On Fermat's equation in integral 2x2 matrices, Period. Math. Hung. 31 (1995), 219-222. Zbl0849.11035
11. [11] Z. Patay and A. Szakacs, On Fermat's problem in matrix rings and groups, Publ. Math. Debrecen 61 (3-4) (2002), 487-494. Zbl1006.11012
12. [12] H. Qin, Fermat’s problem and Goldbach problem over ${M}_{n}\left(Z\right)$, Linear Algebra App., 236 (1996), 131-135.
13. [13] P. Ribenboim, 13 Lectures on Fermat's Last Theorem (New York: Springer-Verlag) 1979. Zbl0456.10006
14. [14] N. Vaserstein, Non-commutative Number Theory, Contemp. Math. 83 (1989), 445-449.

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