{
  "id": "1711.00180",
  "version": "v1",
  "published": "2017-11-01T02:56:50.000Z",
  "updated": "2017-11-01T02:56:50.000Z",
  "title": "Diophantine equations involving Euler's totient function",
  "authors": [
    "Yong-Gao Chen",
    "Hao Tian"
  ],
  "comment": "40 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we consider the equations involving Euler's totient function $\\phi$ and Lucas type sequences. In particular, we prove that the equation $\\phi (x^m-y^m)=x^n-y^n$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a+1, a, 1, 1)$, where $a$ is a positive integer, and the equation $\\phi ((x^m-y^m)/(x-y))=(x^n-y^n)/(x-y)$ has no solutions in positive integers $x, y, m, n$ except for the trivial solutions $(x, y, m , n)=(a, b, 1, 1)$, where $a, b$ are integers with $a>b\\ge 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-01T02:56:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11D61",
      "11D72"
    ],
    "keywords": [
      "eulers totient function",
      "diophantine equations",
      "positive integer",
      "trivial solutions",
      "lucas type sequences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 40,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}