{
  "id": "0905.0635",
  "version": "v19",
  "published": "2009-05-05T18:59:03.000Z",
  "updated": "2014-10-09T15:40:01.000Z",
  "title": "On universal sums of polygonal numbers",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "41 pages, revised version for publication",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For $m=3,4,\\ldots$, the polygonal numbers of order $m$ are given by $p_m(n)=(m-2)\\binom n2+n\\ (n=0,1,2,\\ldots)$. For positive integers $a,b,c$ and $i,j,k\\ge3$ with $\\max\\{i,j,k\\}\\ge5$, we call the triple $(ap_i,bp_j,cp_k)$ universal if for any $n=0,1,2,\\ldots$ there are nonnegative integers $x,y,z$ such that $n=ap_i(x)+bp_j(y)+cp_k(z)$. We show that there are only 95 candidates for universal triples (two of which are $(p_4,p_5,p_6)$ and $(p_3,p_4,p_{27})$), and conjecture that they are indeed universal triples. For many triples $(ap_i,bp_j,cp_k)$ (including $(p_3,4p_4,p_5),(p_4,p_5,p_6)$ and $(p_4,p_4,p_5)$), we prove that any nonnegative integer can be written in the form $ap_i(x)+bp_j(y)+cp_k(z)$ with $x,y,z\\in\\mathbb Z$. We also show some related new results on ternary quadratic forms, one of which states that any nonnegative integer $n\\equiv 1\\pmod{6}$ can be written in the form $x^2+3y^2+24z^2$ with $x,y,z\\in\\mathbb Z$. In addition, we pose several related conjectures one of which states that for any $m=3,4,\\ldots$ each natural number can be expressed as $p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r$ with $x_1,x_2,x_3\\in\\{0,1,2,\\ldots\\}$ and $r\\in\\{0,\\ldots,m-3\\}$.",
  "revisions": [
    {
      "version": "v18",
      "updated": "2011-10-26T15:58:29.000Z",
      "abstract": "For m=3,4,..., the polygonal numbers of order m are given by $p_m(n)=(m-2)n(n-1)/2+n (n=0,1,2,...)$. For positive integers $a,b,c$ and $i,j,k>2$ with max{i,j,k}>4, we call the triple $(ap_i,bp_j,cp_k)$ universal if for any n=0,1,2,... there are nonnegative integers $x,y,z$ such that $n=ap_i(x)+bp_j(y)+cp_k(z)$. We show that there are only 95 candidates for universal triples (two of which are $(p_4,p_5,p_6)$ and $(p_3,p_4,p_{27})$), and conjecture that they are indeed universal triples. For many triples $(ap_i,bp_j,cp_k)$ (including (p_3,4p_4,p_5), (p_4,p_5,p_6) and (p_4,p_4,p_5)), we prove that any nonnegative integer can be written in the form $ap_i(x)+bp_j(y)+cp_k(z)$ with $x,y,z\\in\\Z$. We also show some related new results on ternary quadratic forms, one of which states that any nonnegative integer n=1(mod 6) can be written in the form $x^2+3y^2+24z^2$ with $x,y,z\\in\\Z$. In addition, we pose several related conjectures one of which states that for any m=3,4,... each natural number can be written as p_{m+1}(x_1)+p_{m+2}(x_2)+p_{m+3}(x_3)+r with x_1,x_2,x_3 nonnegative integers and r among 0,...,m-3.",
      "comment": "49 pages, typos corrected",
      "journal": null,
      "doi": null
    },
    {
      "version": "v19",
      "updated": "2014-10-09T15:40:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11E20",
      "11D85",
      "11A41",
      "11P32",
      "11B75"
    ],
    "keywords": [
      "polygonal numbers",
      "universal sums",
      "nonnegative integer",
      "universal triples",
      "ternary quadratic forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0905.0635S"
    }
  }
}