{
  "id": "2002.12155",
  "version": "v1",
  "published": "2020-02-27T15:04:44.000Z",
  "updated": "2020-02-27T15:04:44.000Z",
  "title": "Solutions of $φ(n)=φ(n+k)$ and $σ(n)=σ(n+k)$",
  "authors": [
    "Kevin Ford"
  ],
  "comment": "5 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that for some $k\\le 6990$ and all $k$ with $3099044504245996706400|k$, the equation $\\phi(n)=\\phi(n+k)$ has infinitely many solutions $n$, where $\\phi$ is Euler's totient function. We also show that for a positive proportion of all $k$, the equation $\\sigma(n)=\\sigma(n+k)$ has infinitely many solutions $n$. The proofs rely on recent progress on the prime $k$-tuples conjecture by Zhang, Maynard, Tao and PolyMath.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-27T15:04:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25"
    ],
    "keywords": [
      "eulers totient function",
      "tuples conjecture",
      "positive proportion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}