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On the primitive divisors of the recurrent sequence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1 with applications to group theory. / Vsemirnov, Maxim.

In: Science China Mathematics, Vol. 61, No. 11, 01.11.2018, p. 2101-2110.

Research output: Contribution to journal › Article › peer-review

Harvard

Vsemirnov, M 2018, 'On the primitive divisors of the recurrent sequence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1 with applications to group theory', Science China Mathematics, vol. 61, no. 11, pp. 2101-2110. https://doi.org/10.1007/s11425-017-9347-3

APA

Vsemirnov, M. (2018). On the primitive divisors of the recurrent sequence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1 with applications to group theory. Science China Mathematics, 61(11), 2101-2110. https://doi.org/10.1007/s11425-017-9347-3

Vancouver

Vsemirnov M. On the primitive divisors of the recurrent sequence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1 with applications to group theory. Science China Mathematics. 2018 Nov 1;61(11):2101-2110. https://doi.org/10.1007/s11425-017-9347-3

Author

Vsemirnov, Maxim. / On the primitive divisors of the recurrent sequence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1 with applications to group theory. In: Science China Mathematics. 2018 ; Vol. 61, No. 11. pp. 2101-2110.

BibTeX

@article{a6f7dec014fe4bc1a2582d545b611307,
title = "On the primitive divisors of the recurrent sequence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1 with applications to group theory",
abstract = "Consider the sequence of algebraic integers un given by the starting values u0 = 0, u1 = 1 and the recurrence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1. We prove that for any n ∉ {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in Z[2cos(2π/7)]. As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL2(q) can be generated by a pair x, y with x2= y3= (xy) 7= 1 and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken.",
keywords = "11B37, 20F05, Hurwitz groups, primitive divisors, recurrent sequences, TRIANGLE GROUPS, LUCAS, QUOTIENTS, HURWITZ GROUPS",
author = "Maxim Vsemirnov",
year = "2018",
month = nov,
day = "1",
doi = "10.1007/s11425-017-9347-3",
language = "English",
volume = "61",
pages = "2101--2110",
journal = "Science China Mathematics",
issn = "1674-7283",
publisher = "Science in China Press",
number = "11",

}

RIS

TY - JOUR

T1 - On the primitive divisors of the recurrent sequence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1 with applications to group theory

AU - Vsemirnov, Maxim

PY - 2018/11/1

Y1 - 2018/11/1

N2 - Consider the sequence of algebraic integers un given by the starting values u0 = 0, u1 = 1 and the recurrence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1. We prove that for any n ∉ {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in Z[2cos(2π/7)]. As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL2(q) can be generated by a pair x, y with x2= y3= (xy) 7= 1 and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken.

AB - Consider the sequence of algebraic integers un given by the starting values u0 = 0, u1 = 1 and the recurrence un + 1= (4 cos2(2 π/ 7) − 1) un−un − 1. We prove that for any n ∉ {1, 2, 3, 5, 8, 12, 18, 28, 30} the n-th term of the sequence has a primitive divisor in Z[2cos(2π/7)]. As a consequence we deduce that for any suffciently large n there exists a prime power q such that the group PSL2(q) can be generated by a pair x, y with x2= y3= (xy) 7= 1 and the order of the commutator [x, y] is exactly n. The latter result answers in affrmative a question of Holt and Plesken.

KW - 11B37

KW - 20F05

KW - Hurwitz groups

KW - primitive divisors

KW - recurrent sequences

KW - TRIANGLE GROUPS

KW - LUCAS

KW - QUOTIENTS

KW - HURWITZ GROUPS

UR - http://www.scopus.com/inward/record.url?scp=85052912448&partnerID=8YFLogxK

U2 - 10.1007/s11425-017-9347-3

DO - 10.1007/s11425-017-9347-3

M3 - Article

AN - SCOPUS:85052912448

VL - 61

SP - 2101

EP - 2110

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 11

ER -

ID: 36947097