{
  "id": "0810.3223",
  "version": "v1",
  "published": "2008-10-17T19:03:21.000Z",
  "updated": "2008-10-17T19:03:21.000Z",
  "title": "The critical number of finite abelian groups",
  "authors": [
    "Michael Freeze",
    "Weidong Gao",
    "Alfred Geroldinger"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let G be an additive, finite abelian group. The critical number $\\mathsf{cr}(G)$ of $G$ is the smallest positive integer $\\ell$ such that for every subset $S \\subset G \\setminus \\{0\\}$ with $|S| \\ge \\ell$ the following holds: Every element of $G$ can be written as a nonempty sum of distinct elements from $S$. The critical number was first studied by P. Erd\\H{o}s and H. Heilbronn in 1964, and due to the contributions of many authors the value of $\\mathsf {cr}(G)$ is known for all finite abelian groups $G$ except for $G \\cong \\mathbb{Z}/pq\\mathbb{Z}$ where $p,q$ are primes such that $p+\\lfloor2\\sqrt{p-2}\\rfloor+1<q<2p$. We determine that $\\mathsf {cr}(G)=p+q-2$ for such groups.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-10-17T19:03:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11P70",
      "11B50",
      "11B75"
    ],
    "keywords": [
      "finite abelian group",
      "critical number",
      "smallest positive integer",
      "nonempty sum",
      "distinct elements"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0810.3223F"
    }
  }
}