{
  "id": "1809.08431",
  "version": "v1",
  "published": "2018-09-22T12:44:45.000Z",
  "updated": "2018-09-22T12:44:45.000Z",
  "title": "Irregular primes with respect to Genocchi numbers and Artin's primitive root conjecture",
  "authors": [
    "Su Hu",
    "Min-Soo Kim",
    "Pieter Moree",
    "Min Sha"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we introduce and study a variant of Kummer's notion of (ir)regularity of primes which we call G-irregularity. It is based on Genocchi numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$ is G-irregular if it divides at least one of the integers $G_2,G_4,\\ldots, G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case, G-irregularity is related to the divisibility of some class number. Furthermore, we obtain some results on the distribution of G-irregular primes. In particular, we show that each primitive residue class contains infinitely many G-irregular primes and establish non-trivial lower bounds for their number up to a given bound $x$ as $x$ tends to infinity. As a by-product, we obtain some results on the distribution of primes in arithmetic progressions with a prescribed near-primitive root.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-09-22T12:44:45.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "artins primitive root conjecture",
      "genocchi numbers",
      "g-irregular primes",
      "establish non-trivial lower bounds",
      "primitive residue class contains"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}