{
  "id": "1504.02598",
  "version": "v1",
  "published": "2015-04-10T09:10:38.000Z",
  "updated": "2015-04-10T09:10:38.000Z",
  "title": "Primitive prime divisors and the $n$-th cyclotomic polynomial",
  "authors": [
    "S. \\",
    "P. Glasby",
    "Frank Lübeck",
    "Alice C. Niemeyer",
    "Cheryl E. Praeger"
  ],
  "comment": "13 pages, 5 tables",
  "categories": [
    "math.NT"
  ],
  "abstract": "Primitive prime divisors play an important role in group theory and number theory. We study a certain number theoretic quantity, called $\\Phi^*_n(q)$, which is closely related to the cyclotomic polynomial $\\Phi_n(x)$ and to primitive prime divisors of $q^n-1$. Our definition of $\\Phi^*_n(q)$ is novel, and we prove it is equivalent to the definition given by Hering. Given positive constants $c$ and $k$, we give an algorithm for determining all pairs $(n,q)$ with $\\Phi^*_n(q)\\le cn^k$. This algorithm is used to extend (and correct) a result of Hering which is useful for classifying certain families of subgroups of finite linear groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-10T09:10:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T22",
      "11Y40",
      "20G05"
    ],
    "keywords": [
      "th cyclotomic polynomial",
      "finite linear groups",
      "primitive prime divisors play",
      "number theoretic quantity",
      "important role"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150402598G"
    }
  }
}