{
  "id": "1312.3907",
  "version": "v1",
  "published": "2013-12-13T19:13:39.000Z",
  "updated": "2013-12-13T19:13:39.000Z",
  "title": "Diophantine equations with Euler polynomials",
  "authors": [
    "D. Kreso",
    "Cs. Rakaczki"
  ],
  "comment": "to appear in Acta Arithmetica",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we determine possible decompositions of Euler polynomials $E_k(x)$, i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation $$-1^k +2 ^k - \\cdots + (-1)^{x} x^k=g(y),$$ with $g\\in \\mathbb{Q}[X]$ of degree at least $2$ and $k\\geq 7$, has only finitely many integers solutions $x, y$ unless polynomial $g$ can be decomposed in ways that we list explicitly.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-13T19:13:39.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D41",
      "11B68"
    ],
    "keywords": [
      "diophantine equation",
      "lower degree",
      "functional composition",
      "well-known criterion",
      "writing euler polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.3907K"
    }
  }
}