{
  "id": "1406.5951",
  "version": "v3",
  "published": "2014-06-23T15:51:08.000Z",
  "updated": "2015-04-27T16:44:27.000Z",
  "title": "Consecutive primes and Legendre symbols",
  "authors": [
    "Hao Pan",
    "Zhi-Wei Sun"
  ],
  "comment": "11 pages. Refine the proof of Theorem 1.4",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $m$ be any positive integer and let $\\delta_1,\\delta_2\\in\\{1,-1\\}$. We show that for some constanst $C_m>0$ there are infinitely many integers $n>1$ with $p_{n+m}-p_n\\le C_m$ such that $$\\left(\\frac{p_{n+i}}{p_{n+j}}\\right)=\\delta_1\\ \\quad\\text{and}\\ \\quad\\left(\\frac{p_{n+j}}{p_{n+i}}\\right)=\\delta_2$$ for all $0\\le i<j\\le m$, where $p_k$ denotes the $k$-th prime, and $(\\frac {\\cdot}p)$ denotes the Legendre symbol for any odd prime $p$. We also prove that under the Generalized Riemann Hypothesis there are infinitely many positive integers $n$ such that $p_{n+i}$ is a primitive root modulo $p_{n+j}$ for any distinct $i$ and $j$ among $0,1,\\ldots,m$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-26T15:54:22.000Z",
      "comment": "11 pages, minor revision",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-27T16:44:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A41",
      "11A07",
      "11A15",
      "11N99"
    ],
    "keywords": [
      "legendre symbol",
      "consecutive primes",
      "positive integer",
      "th prime",
      "odd prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.5951P"
    }
  }
}