{
  "id": "1704.00151",
  "version": "v1",
  "published": "2017-04-01T10:26:51.000Z",
  "updated": "2017-04-01T10:26:51.000Z",
  "title": "On the greatest common divisor of $n$ and the $n$th Fibonacci number",
  "authors": [
    "Paolo Leonetti",
    "Carlo Sanna"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\mathcal{A}$ be the set of all integers of the form $\\gcd(n, F_n)$, where $n$ is a positive integer and $F_n$ denotes the $n$th Fibonacci number. We prove that $\\#\\left(\\mathcal{A} \\cap [1, x]\\right) \\gg x / \\log x$ for all $x \\geq 2$, and that $\\mathcal{A}$ has zero asymptotic density. Our proofs rely on a recent result of Cubre and Rouse which gives, for each positive integer $n$, an explicit formula for the density of primes $p$ such that $n$ divides the rank of appearance of $p$, that is, the smallest positive integer $k$ such that $p$ divides $F_k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-04-01T10:26:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11A05",
      "11N25"
    ],
    "keywords": [
      "th fibonacci number",
      "greatest common divisor",
      "zero asymptotic density",
      "smallest positive integer",
      "explicit formula"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}