{
  "id": "1406.6851",
  "version": "v1",
  "published": "2014-06-26T11:40:47.000Z",
  "updated": "2014-06-26T11:40:47.000Z",
  "title": "On Primitive Covering Numbers",
  "authors": [
    "Lenny Jones",
    "Daniel White"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "In 2007, Zhi-Wei Sun defined a \\emph{covering number} to be a positive integer $L$ such that there exists a covering system of the integers where the moduli are distinct divisors of $L$ greater than 1. A covering number $L$ is called \\emph{primitive} if no proper divisor of $L$ is a covering number. Sun constructed an infinite set $\\mathcal L$ of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given $L\\in \\mathcal L$, we derive a formula that gives the exact number of coverings that have $L$ as the least common multiple of the set $M$ of moduli, under certain restrictions on $M$. Additionally, we disprove Sun's conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-06-26T11:40:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "infinite set",
      "disprove suns conjecture",
      "primitive covering number condition",
      "distinct divisors",
      "proper divisor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1406.6851J"
    }
  }
}