{
  "id": "1406.4801",
  "version": "v2",
  "published": "2014-06-03T04:01:42.000Z",
  "updated": "2014-06-19T19:46:05.000Z",
  "title": "Two statements that are equivalent to a conjecture related to the distribution of prime numbers",
  "authors": [
    "Germán Paz"
  ],
  "comment": "16 pages, 3 figures (version 2 includes them also as ancillary files, no changes to version 1 have been made), Mathematica code; keywords: Andrica's conjecture, Brocard's conjecture, Legendre's conjecture, Oppermann's conjecture, prime numbers, triangular numbers. arXiv admin note: text overlap with arXiv:1310.1323",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $n\\in\\mathbb{Z}^+$. In [8] we ask the question whether any sequence of $n$ consecutive integers greater than $n^2$ and smaller than $(n+1)^2$ contains at least one prime number, and we show that this is actually the case for every $n\\leq 1,193,806,023$. In addition, we prove that a positive answer to the previous question for all $n$ would imply Legendre's, Brocard's, Andrica's, and Oppermann's conjectures, as well as the assumption that for every $n$ there is always a prime number in the interval $[n,n+2\\lfloor\\sqrt{n}\\rfloor-1]$. Let $\\pi[n+g(n),n+f(n)+g(n)]$ denote the amount of prime numbers in the interval $[n+g(n),n+f(n)+g(n)]$. Here we show that the conjecture described in [8] is equivalent to the statement that $$\\pi[n+g(n),n+f(n)+g(n)]\\ge 1\\text{, }\\forall n\\in\\mathbb{Z}^+\\text{,}$$ where $$f(n)=\\left(\\frac{n-\\lfloor\\sqrt{n}\\rfloor^2-\\lfloor\\sqrt{n}\\rfloor-\\beta}{|n-\\lfloor\\sqrt{n}\\rfloor^2-\\lfloor\\sqrt{n}\\rfloor-\\beta|}\\right)(1-\\lfloor\\sqrt{n}\\rfloor)\\text{, }g(n)=\\left\\lfloor1-\\sqrt{n}+\\lfloor\\sqrt{n}\\rfloor\\right\\rfloor\\text{,}$$ and $\\beta$ is any real number such that $1<\\beta<2$. We also prove that the conjecture in question is equivalent to the statement that $$\\pi[S_n,S_n+\\lfloor\\sqrt{S_n}\\rfloor-1]\\ge 1\\text{, }\\forall n\\in\\mathbb{Z}^+\\text{,}$$ where $$S_n=n+\\frac{1}{2}\\left\\lfloor\\frac{\\sqrt{8n+1}-1}{2}\\right\\rfloor^2-\\frac{1}{2}\\left\\lfloor\\frac{\\sqrt{8n+1}-1}{2}\\right\\rfloor+1\\text{.}$$ We use this last result in order to create plots of $h(n)=\\pi[S_n,S_n+\\lfloor\\sqrt{S_n}\\rfloor-1]$ for many values of $n$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-06-19T19:46:05.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "00A05",
      "11A41"
    ],
    "keywords": [
      "prime number",
      "equivalent",
      "conjecture",
      "distribution",
      "consecutive integers greater"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.4801P"
    }
  }
}