{
  "id": "1211.7206",
  "version": "v1",
  "published": "2012-11-30T10:01:30.000Z",
  "updated": "2012-11-30T10:01:30.000Z",
  "title": "A Conjecture Connected with Units of Quadratic Fields",
  "authors": [
    "Nihal Bircan"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In this article, we consider the order $\\mathcal{O}_{f}={x+yf\\sqrt{d}:x,\\ y \\in \\Z}$ with conductor $f\\in\\N$ in a real quadratic field $K=\\mathbb{Q}(\\sqrt{d})$ where $d>0$ is square-free and $d\\equiv2,3\\pmod 4$. We obtain numerical information about $ n(f)=n(p)=min{\\nu\\in\\N : \\varepsilon^{\\nu}\\in \\mathcal{O}_{p}}$ where $\\varepsilon>1$ is the fundamental unit of $K$ and $p$ is an odd prime. Our numerical results suggest that the frequencies of $\\frac{p\\pm1}{2n(p)}$ or $\\frac{p\\pm1}{n(p)}$ should have a limit as the ranges of $d$ and $p$ go to infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-11-30T10:01:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11Y40",
      "11R27",
      "11R11"
    ],
    "keywords": [
      "conjecture",
      "real quadratic field",
      "fundamental unit",
      "odd prime",
      "square-free"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1211.7206B"
    }
  }
}