On the matrix negative Pell equation

Aleksander Grytczuk; Izabela Kurzydło

Discussiones Mathematicae - General Algebra and Applications (2009)

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

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) ∑ i = 1 n X ₂ i - d ∑ i = 1 n Y ² i = - I with d ∈ N for nonsingular X i , Y i ∈ M ₂ ( Z ) , 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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  2. [2] R.Z. Domiaty, Solutions of x⁴+y⁴=z⁴ in 2x2 integral matrices, Amer. Math. Monthly (1966) 73, 631. 
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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 ( Z ) , Linear Algebra App., 236 (1996), 131-135. 
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  14. [14] N. Vaserstein, Non-commutative Number Theory, Contemp. Math. 83 (1989), 445-449. 

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