### Abstract

Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called φ∗
_{n}(q), which is closely related to the cyclotomic polynomial φ
_{n} (x) and to primitive prime divisors of q
^{n} - 1. Our definition of φ∗
_{n}(q) is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants c and k, we provide an algorithm for determining all pairs (n; q) with φ∗
_{n}(q) ≤ cn
^{k}. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

Original language | English |
---|---|

Pages (from-to) | 122-135 |

Number of pages | 14 |

Journal | Journal of the Australian Mathematical Society |

Volume | 102 |

Issue number | 1 |

DOIs | |

Publication status | Published - 2017 |

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### Cite this

*Journal of the Australian Mathematical Society*,

*102*(1), 122-135. https://doi.org/10.1017/S1446788715000269

}

*Journal of the Australian Mathematical Society*, vol. 102, no. 1, pp. 122-135. https://doi.org/10.1017/S1446788715000269

**Primitive prime divisors and the nTH cyclotomic polynomial.** / GLASBY, Stephen; Lubeck, Frank; Niemeyer, Alice; Praeger, Cheryl.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Primitive prime divisors and the nTH cyclotomic polynomial

AU - GLASBY, Stephen

AU - Lubeck, Frank

AU - Niemeyer, Alice

AU - Praeger, Cheryl

PY - 2017

Y1 - 2017

N2 - Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called φ∗ n(q), which is closely related to the cyclotomic polynomial φ n (x) and to primitive prime divisors of q n - 1. Our definition of φ∗ n(q) is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants c and k, we provide an algorithm for determining all pairs (n; q) with φ∗ n(q) ≤ cn k. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

AB - Primitive prime divisors play an important role in group theory and number theory. We study a certain number-theoretic quantity, called φ∗ n(q), which is closely related to the cyclotomic polynomial φ n (x) and to primitive prime divisors of q n - 1. Our definition of φ∗ n(q) is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants c and k, we provide an algorithm for determining all pairs (n; q) with φ∗ n(q) ≤ cn k. This algorithm is used to extend (and correct) a result of Hering and is useful for classifying certain families of subgroups of finite linear groups.

KW - Zsigmondy primes

KW - cyclotomic polynomial

UR - http://www.scopus.com/inward/record.url?scp=84946108636&partnerID=8YFLogxK

UR - http://www.mendeley.com/research/primitive-prime-divisors-nth-cyclotomic-polynomial

U2 - 10.1017/S1446788715000269

DO - 10.1017/S1446788715000269

M3 - Article

VL - 102

SP - 122

EP - 135

JO - Journal of the Australian Mathematical Society Series A-Pure Mathematics and Statistics

JF - Journal of the Australian Mathematical Society Series A-Pure Mathematics and Statistics

SN - 1446-7887

IS - 1

ER -