{
  "id": "2109.09128",
  "version": "v1",
  "published": "2021-09-19T14:10:28.000Z",
  "updated": "2021-09-19T14:10:28.000Z",
  "title": "Differences between perfect powers : the Lebesgue-Nagell Equation",
  "authors": [
    "Michael A. Bennett",
    "Samir Siksek"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We develop a variety of new techniques to treat Diophantine equations of the shape $x^2+D =y^n$, based upon bounds for linear forms in $p$-adic and complex logarithms, the modularity of Galois representations attached to Frey-Hellegouarch elliptic curves, and machinery from Diophantine approximation. We use these to explicitly determine the set of all coprime integers $x$ and $y$, and $n \\geq 3$, with the property that $y^n > x^2$ and $x^2-y^n$ has no prime divisor exceeding $11$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-09-19T14:10:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D61"
    ],
    "keywords": [
      "lebesgue-nagell equation",
      "perfect powers",
      "differences",
      "treat diophantine equations",
      "frey-hellegouarch elliptic curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}