{
  "id": "1707.03317",
  "version": "v1",
  "published": "2017-07-11T15:15:29.000Z",
  "updated": "2017-07-11T15:15:29.000Z",
  "title": "The rational part of a periodic continued fraction",
  "authors": [
    "Kurt Girstmair"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $x$ be a periodic continued fraction with the initial block $0$ and the repeating block $c_1,\\ldots,c_n$. So $x$ is a quadratic irrational of the form $x=a+\\sqrt b$, where $a$, $b$ are rational numbers, $b>0$, $b$ not a square. The numbers $a$ and $\\sqrt b$ are uniquely determined by $x$. In general it is difficult to say what the influence of a certain digit of the repeating block on the appearance of $x$ is. We highlight a noteworthy exception from this rule. Indeed, the magnitude of $2a$ is essentially determined by the last digit $c_n$ of the repeating block, the fractional part of $2a$, however, is independent of $c_n$. Of particular interest is the case $2a\\in\\mathbb Z$, which occurs if, and only if, the sequence $c_1,\\ldots,c_{n-1}$ is symmetric.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-11T15:15:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A55"
    ],
    "keywords": [
      "periodic continued fraction",
      "rational part",
      "repeating block",
      "quadratic irrational",
      "rational numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}