{
  "id": "1612.01468",
  "version": "v1",
  "published": "2016-12-05T18:45:06.000Z",
  "updated": "2016-12-05T18:45:06.000Z",
  "title": "Consecutive primes and Beatty sequences",
  "authors": [
    "William D. Banks",
    "Victor Z. Guo"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix irrational numbers $\\alpha,\\hat\\alpha>1$ of finite type and real numbers $\\beta,\\hat\\beta\\ge 0$, and let $B$ and $\\hat B$ be the Beatty sequences $$ B:=(\\lfloor\\alpha m+\\beta\\rfloor)_{m\\ge 1}\\quad\\text{and}\\quad\\hat B:=(\\lfloor\\hat\\alpha m+\\hat\\beta\\rfloor)_{m\\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\\sharp)$ of consecutive primes for which $p\\in B$ and $p^\\sharp\\in\\hat B$. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that $$ \\big|\\{p\\le x:p\\in B\\text{ and }p^\\sharp\\in\\hat B\\}\\big|=(\\alpha\\hat\\alpha)^{-1}\\pi(x)+O\\big(x(\\log x)^{-3/2+\\epsilon}\\big), $$ where $\\pi(x)$ is the prime counting function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-05T18:45:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N05",
      "11B83"
    ],
    "keywords": [
      "beatty sequences",
      "consecutive primes",
      "fix irrational numbers",
      "finite type",
      "prime counting function"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}