{
  "id": "1502.03056",
  "version": "v1",
  "published": "2015-02-09T16:57:07.000Z",
  "updated": "2015-02-09T16:57:07.000Z",
  "title": "On universal sums $ax^2+by^2+f(z)$, $aT_x+bT_y+f(z)$ and $aT_x+by^2+f(z)$",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "For integer-valued polynomials $f_1(x),f_2(y),f_3(z)$, if any $n\\in\\mathbb N=\\{0,1,2,...\\}$ can be written as $f_1(x)+f_2(y)+f_3(z)$ with $x,y,z\\in\\mathbb N$ then we say that $f_1(x)+f_2(y)+f_3(z)$ is universal (over $\\mathbb N$). In this paper we find all candidates of universal sums of the following three types: $$ax^2+by^2+f(z),\\ aT_x+bT_y+f(z),\\ aT_x+by^2+f(z),$$ where $T_x$ denotes $x(x+1)/2$ and $f(z)$ has the form $c\\binom z2+dz$ with $c,d\\in\\{1,2,3,...\\}$ and $d\\nmid c$. We also show that some of the candidates (including $T_x+y^2+z(z+2k)$ for $k=1,2,3$, and $T_x+y^2+z(z+2k+1)/2$ for $k=1,...,7$) are indeed universal sums.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-09T16:57:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11B75",
      "11D85",
      "11E20",
      "11P32"
    ],
    "keywords": [
      "universal sums",
      "candidates",
      "integer-valued polynomials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}