{
  "id": "2003.12749",
  "version": "v1",
  "published": "2020-03-28T09:13:53.000Z",
  "updated": "2020-03-28T09:13:53.000Z",
  "title": "On the exponential Diophantine equation $(n-1)^{x}+(n+2)^{y}=n^{z}$",
  "authors": [
    "Hairong Bai",
    "Elif Kızıldere",
    "Gökhan Soydan",
    "Pingzhi Yuan"
  ],
  "comment": "12 pages, to appear, Colloquium Mathematicum (2020)",
  "doi": "10.4064/cm7668-6-2019",
  "categories": [
    "math.NT"
  ],
  "abstract": "Suppose that $n$ is a positive integer. In this paper, we show that the exponential Diophantine equation $$(n-1)^{x}+(n+2)^{y}=n^{z},\\ n\\geq 2,\\ xyz\\neq 0$$ has only the positive integer solutions $(n,x,y,z)=(3,2,1,2), (3,1,2,3)$. The main tools on the proofs are Baker's theory and Bilu-Hanrot-Voutier's result on primitive divisors of Lucas numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-03-28T09:13:53.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61",
      "11D41"
    ],
    "keywords": [
      "exponential diophantine equation",
      "main tools",
      "bilu-hanrot-voutiers result",
      "positive integer solutions",
      "bakers theory"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}