{
  "id": "1309.0479",
  "version": "v1",
  "published": "2013-09-02T18:24:22.000Z",
  "updated": "2013-09-02T18:24:22.000Z",
  "title": "On the Interval [n,2n]: Primes, Composites and Perfect Powers",
  "authors": [
    "Germán Paz"
  ],
  "comment": "15 pages, 2 tables",
  "journal": "General Mathematics Notes, Vol. 15, No. 1, March 2013, pp. 1-15",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we show that for every positive integer $n$ there exists a prime number in the interval $[n,9(n+3)/8]$. Based on this result, we prove that if $a$ is an integer greater than 1, then for every integer $n>14.4a$ there are at least four prime numbers $p$, $q$, $r$, and $s$ such that $n<ap<3n/2<aq<2n$ and $n<r<3n/2<s<2n$. Moreover, we also prove that if $m$ is a positive integer, then for every positive integer $n>14.4/(|\\sqrt[m]{1.5}|-1)^m$ there exist a positive integer $a$ and a prime number $s$ such that $n<a^m<3n/2<s<2n$, as well as the fact that for every positive integer $n>14.4/(|\\sqrt[m]{2}|-|\\sqrt[m]{1.5}|)^m$ there exist a prime number $r$ and a positive integer $a$ such that $n<r<3n/2<a^m<2n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-09-02T18:24:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "00A05",
      "11A41"
    ],
    "keywords": [
      "positive integer",
      "prime number",
      "perfect powers",
      "composites"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1309.0479P"
    }
  }
}