{
  "id": "1709.00400",
  "version": "v1",
  "published": "2017-09-01T17:40:50.000Z",
  "updated": "2017-09-01T17:40:50.000Z",
  "title": "On the Diophantine equation $(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}=y^n$",
  "authors": [
    "Attila Bérczes",
    "István Pink",
    "Gamze SavaŞ",
    "Gökhan Soydan"
  ],
  "comment": "26 pages, accepted for publication in Journal of Number Theory (2017)",
  "doi": "10.1016/j.jnt.2017.07.020",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this work, we give upper bounds for $n$ on the title equation. Our results depend on assertions describing the precise exponents of $2$ and $3$ appearing in the prime factorization of $T_{k}(x)=(x+1)^{k}+(x+2)^{k}+...+(2x)^{k}$. Further, on combining Baker's method with the explicit solution of polynomial exponential congruences (see e.g. BHMP), we show that for $2 \\leq x \\leq 13, k \\geq 1, y \\geq 2$ and $n \\geq 3$ the title equation has no solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-01T17:40:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11D61"
    ],
    "keywords": [
      "diophantine equation",
      "title equation",
      "polynomial exponential congruences",
      "prime factorization",
      "upper bounds"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}