{
  "id": "0903.1742",
  "version": "v1",
  "published": "2009-03-10T11:59:34.000Z",
  "updated": "2009-03-10T11:59:34.000Z",
  "title": "The Diophantine equation $aX^{4} - bY^{2} = 1$",
  "authors": [
    "Shabnam Akhtari"
  ],
  "comment": "20 pages, To appear in Journal fur die Reine und Angewandte Mathematik (Crelle's Journal)",
  "categories": [
    "math.NT"
  ],
  "abstract": "As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation $aX^{4} - bY^2=1$, for fixed positive integers $a$ and $b$, possesses at most two solutions in positive integers $X$ and $Y$. Since there are infinitely many pairs $(a,b)$ for which two such solutions exist, this result is sharp.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-03-10T11:59:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D25",
      "11D41",
      "11B39",
      "11J25"
    ],
    "keywords": [
      "diophantine equation",
      "conjecture",
      "fixed positive integers",
      "application"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0903.1742A"
    }
  }
}