{
  "id": "1407.7186",
  "version": "v1",
  "published": "2014-07-27T05:26:00.000Z",
  "updated": "2014-07-27T05:26:00.000Z",
  "title": "Bounded gaps between primes with a given primitive root, II",
  "authors": [
    "Roger C. Baker",
    "Paul Pollack"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $m$ be a natural number, and let $\\mathcal{Q}$ be a set containing at least $\\exp(C m)$ primes. We show that one can find infinitely many strings of $m$ consecutive primes each of which has some $q\\in\\mathcal{Q}$ as a primitive root, all lying in an interval of length $O_{\\mathcal{Q}}(\\exp(C'm))$. This is a bounded gaps variant of a theorem of Gupta and Ram Murty. We also prove a result on an elliptic analogue of Artin's conjecture. Let $E/\\mathbb{Q}$ be an elliptic curve with an irrational $2$-torsion point. Assume GRH. Then for every $m$, there are infinitely many strings of $m$ consecutive primes $p$ for which $E(\\mathbb{F}_p)$ is cyclic, all lying an interval of length $O_E(\\exp(C'' m))$. If $E$ has CM, then the GRH assumption can be removed. Here $C$, $C'$, and $C''$ are absolute constants.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-27T05:26:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11G05"
    ],
    "keywords": [
      "primitive root",
      "consecutive primes",
      "absolute constants",
      "bounded gaps variant",
      "natural number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.7186B"
    }
  }
}