{
  "id": "2111.06963",
  "version": "v2",
  "published": "2021-11-05T02:40:19.000Z",
  "updated": "2023-10-18T15:34:15.000Z",
  "title": "Bertrand's Postulate for Carmichael Numbers",
  "authors": [
    "Daniel Larsen"
  ],
  "comment": "27 pages; corrected minor issues",
  "journal": "Int. Math. Res. Not. (2023), no. 15, 13072-13098",
  "categories": [
    "math.NT"
  ],
  "abstract": "Alford, Granville, and Pomerance proved that there are infinitely many Carmichael numbers. In the same paper, they ask if a statement analogous to Bertrand's postulate could be proven for Carmichael numbers. In this paper, we answer this question, proving the stronger statement that for all $\\delta>0$ and $x$ sufficiently large in terms of $\\delta$, there exist at least $e^{\\frac{\\log x}{(\\log\\log x)^{2+\\delta}}}$ Carmichael numbers between $x$ and $x+\\frac{x}{(\\log x)^{\\frac{1}{2+\\delta}}}$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2023-10-18T15:34:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11N25"
    ],
    "keywords": [
      "carmichael numbers",
      "bertrands postulate",
      "stronger statement",
      "sufficiently large"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}