{
  "id": "1511.06177",
  "version": "v1",
  "published": "2015-11-19T14:15:42.000Z",
  "updated": "2015-11-19T14:15:42.000Z",
  "title": "Some relations between t(a,b,c,d;n) and N(a,b,c,d;n)",
  "authors": [
    "Zhi-Hong Sun"
  ],
  "comment": "11 pages. arXiv admin note: substantial text overlap with arXiv:1511.00478",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $\\Bbb Z$ and $\\Bbb N$ be the set of integers and the set of positive integers, respectively. For $a,b,c,d,n\\in\\Bbb N$ let $N(a,b,c,d;n)$ be the number of representations of $n$ by $ax^2+by^2+cz^2+dw^2$, and let $t(a,b,c,d;n)$ be the number of representations of $n$ by $ax(x-1)/2+by(y-1)/2+cz(z-1)/2 +dw(w-1)/2$ $(x,y,z,w\\in\\Bbb Z$). In this paper we reveal some connections between $t(a,b,c,d;n)$ and $N(a,b,c,d;n)$. Suppose $a,n\\in\\Bbb N$ and $a\\equiv 1\\pmod 2$. We show that $$t(a,b,c,d;n)=\\frac 23(N(a,b,c,d;8n+a+b+c+d)-N(a,b,c,d;2n+(a+b+c+d)/4))$$ for $(a,b,c,d)= (a,a,2a,8m+4)$ and $(a,3a,4k+2,4m+2)$ with $k\\equiv m\\pmod 2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-19T14:15:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D85",
      "11E25"
    ],
    "keywords": [
      "representations",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151106177S"
    }
  }
}