{
  "id": "1608.02022",
  "version": "v1",
  "published": "2016-08-08T15:32:59.000Z",
  "updated": "2016-08-08T15:32:59.000Z",
  "title": "Sums of four polygonal numbers with coefficients",
  "authors": [
    "Xiang-Zi Meng",
    "Zhi-Wei Sun"
  ],
  "comment": "22 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $m\\ge3$ be a positive integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\\binom n2+n$ $(n=0,1,2,\\ldots)$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1) + p_{m+2}(x_2) + ap_{m+2}(x_3) + bp_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\\in\\{(1,3),(2,4)\\}$, or $(a,b)=(1,1)\\ \\&\\ 4\\mid m$, or $(a,b)=(2,2)\\ \\&\\ m\\not\\equiv 2\\pmod 4$. In particular, we confirm a conjecture of Z.-W. Sun which states that any natural number can be written as $p_6(x_1) + p_6(x_2) + 2p_6(x_3) + 4p_6(x_4)$ with $x_1,x_2,x_3,x_4$ nonnegative integers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-08T15:32:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E20",
      "11E25",
      "11B13",
      "11B75",
      "11D85",
      "11P99"
    ],
    "keywords": [
      "polygonal numbers",
      "coefficients",
      "nonnegative integers",
      "natural number",
      "sufficiently large integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}