{
  "id": "0710.1936",
  "version": "v3",
  "published": "2007-10-10T06:51:33.000Z",
  "updated": "2008-09-01T07:59:54.000Z",
  "title": "Regular integers modulo n",
  "authors": [
    "László Tóth"
  ],
  "comment": "9 pages, final version",
  "journal": "Annales Univ. Sci. Budapest., Sect. Comp., 29 (2008), 263-275",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $n=p_1^{\\nu_1}... p_r^{\\nu_r} >1$ be an integer. An integer $a$ is called regular (mod $n$) if there is an integer $x$ such that $a^2x\\equiv a$ (mod $n$). Let $\\varrho(n)$ denote the number of regular integers $a$ (mod $n$) such that $1\\le a\\le n$. Here $\\varrho(n)=(\\phi(p_1^{\\nu_1})+1)... (\\phi(p_r^{\\nu_r})+1)$, where $\\phi(n)$ is the Euler function. In this paper we first summarize some basic properties of regular integers (mod $n$). Then in order to compare the rates of growth of the functions $\\varrho(n)$ and $\\phi(n)$ we investigate the average orders and the extremal orders of the functions $\\varrho(n)/\\phi(n)$, $\\phi(n)/\\varrho(n)$ and $1/\\varrho(n)$.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-09-01T07:59:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A25",
      "11N37"
    ],
    "keywords": [
      "regular integers modulo",
      "basic properties",
      "euler function",
      "average orders",
      "extremal orders"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0710.1936T"
    }
  }
}