{
  "id": "2211.12160",
  "version": "v1",
  "published": "2022-11-22T10:40:37.000Z",
  "updated": "2022-11-22T10:40:37.000Z",
  "title": "Units from square-roots of rational numbers",
  "authors": [
    "Kurt Girstmair"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $D,Q$ be natural numbers, $(D,Q)=1$, such that $D/Q>1$ and $D/Q$ is not a square. Let $q$ be the smallest divisor of $Q$ such that $Q|\\, q^2$. We show that the units $>1$ of the ring $\\mathbb Z[\\sqrt{Dq^2/Q}]$ are connected with certain convergents of $\\sqrt{D/Q}$. Among these units, the units of $\\mathbb Z[\\sqrt{DQ}]$ play a special role, inasmuch as they correspond to the convergents of $\\sqrt{D/Q}$ that occur just before the end of each period. We also show that the last-mentioned units allow reading the (periodic) continued fraction expansion of certain quadratic irrationals from the (finite) continued fraction expansion of certain rational numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-22T10:40:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55",
      "11R11"
    ],
    "keywords": [
      "rational numbers",
      "continued fraction expansion",
      "square-roots",
      "natural numbers",
      "smallest divisor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}