{
  "id": "1407.3261",
  "version": "v2",
  "published": "2014-07-11T19:45:31.000Z",
  "updated": "2014-08-06T02:38:42.000Z",
  "title": "Proof of a conjecture of Guy on class numbers",
  "authors": [
    "Lynn Chua",
    "Benjamin Gunby",
    "Soohyun Park",
    "Allen Yuan"
  ],
  "comment": "9 pages; additional background given in introduction concerning $h(p)$ and $h(-p)$ modulo small powers of 2",
  "categories": [
    "math.NT"
  ],
  "abstract": "It is well known that for any prime $p\\equiv 3$ (mod $4$), the class numbers of the quadratic fields $\\mathbb{Q}(\\sqrt{p})$ and $\\mathbb{Q}(\\sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a formula for $h(p)/h(-p)$ modulo powers of $2$. We show the formula $h(p) \\equiv h(-p) m(p)$ (mod $16$), where $m(p)$ is an integer defined using the \"negative\" continued fraction expansion of $\\sqrt{p}$. Our result solves a conjecture of Richard Guy.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-08-06T02:38:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R29"
    ],
    "keywords": [
      "class numbers",
      "conjecture",
      "continued fraction expansion",
      "quadratic fields",
      "modulo powers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3261C"
    }
  }
}