TY - JOUR
T1 - Some remarks on fermat’s equation in the set of matrices
AU - Cao, Zhenfu
AU - Grytczuk, Aleksander
N1 - Publisher Copyright:
© 2000, Eszterhazy Karoly College. All rights reserved.
PY - 2000
Y1 - 2000
N2 - Let Z be the set of integers and SL2(Z) the set of 2×2 integral matrices with det A=1 for A∈SL2(Z). If any two of SL2(Z) are commutative, then the set of such matrices we denote by SL2(Z). In this paper, we prove that Fermat’s equation (*) Xn+Yn=Zn has a solution in the set SL2(Z) if and only if n≡1 (mod 6) or n≡5 (mod 6). This criterion is connected with a criterion given recently by Khazanov [4]. Moreover, we indicate a subclass of the matrices of SL2(Z) for which (*) has no solutions for arbitrary positive integers n≥2.
AB - Let Z be the set of integers and SL2(Z) the set of 2×2 integral matrices with det A=1 for A∈SL2(Z). If any two of SL2(Z) are commutative, then the set of such matrices we denote by SL2(Z). In this paper, we prove that Fermat’s equation (*) Xn+Yn=Zn has a solution in the set SL2(Z) if and only if n≡1 (mod 6) or n≡5 (mod 6). This criterion is connected with a criterion given recently by Khazanov [4]. Moreover, we indicate a subclass of the matrices of SL2(Z) for which (*) has no solutions for arbitrary positive integers n≥2.
UR - https://www.scopus.com/pages/publications/85042133649
M3 - 文章
AN - SCOPUS:85042133649
SN - 1787-5021
VL - 27
SP - 47
EP - 52
JO - Annales Mathematicae et Informaticae
JF - Annales Mathematicae et Informaticae
ER -