{
  "id": "1211.4571",
  "version": "v1",
  "published": "2012-11-17T18:31:10.000Z",
  "updated": "2012-11-17T18:31:10.000Z",
  "title": "An elemetary proof of an estimate for a number of primes less than the product of the first $n$ primes",
  "authors": [
    "Romeo Meštrović"
  ],
  "comment": "9 pages; we prove a Bonse-type inequality which yields that for each $1<\\alpha <2$ there exist at least $\\[n^\\alpha\\]$ primes which are less than the product of the first $n_0=n_0(\\alpha)$ primes",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\alpha$ be a real number such that $1< \\alpha <2$ and let $x_0=x_0(\\alpha)$ be a {\\rm(}unique{\\rm)} positive solution of the equation $$ x^{\\alpha-1} -\\frac{\\pi}{e^2\\sqrt{3}}x +1=0. $$ Then we prove that for each positive integer $n>x_0$ there exist at least $[n^\\alpha]$ primes between the $(n+1)$th prime and the product of the first $n+1$ primes. In particular, we establish a recent Cooke's result which asserts that for each positive integer $n$ there are at least $n$ primes between the $(n+1)$th prime and the product of the first $n+1$ primes. Our proof is based on an elementary counting method (enumerative arguments) and the application of Stirling's formula to give upper bound for some binomial coefficients.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-17T18:31:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A41",
      "11A51",
      "11A25"
    ],
    "keywords": [
      "elemetary proof",
      "th prime",
      "positive integer",
      "stirlings formula",
      "elementary counting method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.4571M"
    }
  }
}