{
  "id": "0906.2450",
  "version": "v2",
  "published": "2009-06-15T03:47:06.000Z",
  "updated": "2016-02-25T16:07:19.000Z",
  "title": "On some universal sums of generalized polygonal numbers",
  "authors": [
    "Fan Ge",
    "Zhi-Wei Sun"
  ],
  "comment": "7 pages, accepted version for publication in Colloq. Math",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For $m=3,4,\\ldots$ those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\\in\\mathbb Z$ are called generalized $m$-gonal numbers. Sun [S15] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\\mathbb Z$ (i.e., any $n\\in\\mathbb N=\\{0,1,2,\\ldots\\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\\in\\mathbb Z$). In this paper we prove that $p_5+bp_5+3p_5\\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\\in\\mathbb N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\\in\\mathbb N$; in contrast we show that $p_3+p_5+p_{11}$ and $3p_3+p_5+p_7$ are universal over $\\mathbb Z$. Our proofs are essentially elementary and hence suitable for general readers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-06-15T03:47:06.000Z",
      "abstract": "For m=3,4,... those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\\in\\Z$ are called generalized $m$-gonal numbers. Recently the second author studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\\Z$ (i.e., any $n\\in\\N=\\{0,1,2,...\\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\\in\\Z$). In this paper we proved that $p_5+bp_5+3p_5 (b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\\Z$; this partially confirms Sun's conjecture on $ap_5+bp_5+cp_5$. Sun also conjectured that any $n\\in\\N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\\in\\N$; in contrast we show that $p_3+p_5+p_{11}$ and $3p_3+p_5+p_7$ are universal over $\\Z$.",
      "comment": "7 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2016-02-25T16:07:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11D85",
      "11E20",
      "11P32"
    ],
    "keywords": [
      "generalized polygonal numbers",
      "universal sums",
      "partially confirms suns conjecture",
      "second author",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0906.2450G"
    }
  }
}