{
  "id": "2007.05771",
  "version": "v1",
  "published": "2020-07-11T13:32:32.000Z",
  "updated": "2020-07-11T13:32:32.000Z",
  "title": "Gaps between totients",
  "authors": [
    "Kevin Ford",
    "Sergei Konyagin"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study the set D of positive integers d for which the equation $\\phi(a)-\\phi(b)=d$ has infinitely many solution pairs (a,b), where $\\phi$ is Euler's totient function. We show that the minumum of D is at most 154, exhibit a specific A so that every multiple of A is in D, and show that any progression a mod d with 4|a and 4|d, contains infinitely many elements of D. We also show that the Generalized Elliott-Halberstam Conjecture, as defined in [6], implies that D equals the set of all positive, even integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-11T13:32:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N64"
    ],
    "keywords": [
      "eulers totient function",
      "solution pairs",
      "generalized elliott-halberstam conjecture",
      "positive integers",
      "progression"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}