{
  "id": "1707.06223",
  "version": "v1",
  "published": "2017-07-19T17:59:22.000Z",
  "updated": "2017-07-19T17:59:22.000Z",
  "title": "Some universal quadratic sums over the integers",
  "authors": [
    "Hai-Liang Wu",
    "Zhi-Wei Sun"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $a,b,c,d,e,f\\in\\mathbb N$ with $a\\ge c\\ge e>0$, $b\\le a$ and $b\\equiv a\\pmod2$, $d\\le c$ and $d\\equiv c\\pmod2$, $f\\le e$ and $f\\equiv e\\pmod2$. If any nonnegative integer can be written as $x(ax+b)/2+y(cy+d)/2+z(ez+f)/2$ with $x,y,z\\in\\mathbb Z$, then the tuple $(a,b,c,d,e,f)$ is said to be universal over $\\mathbb Z$. Recently, Z.-W. Sun found all candidates of such universal tuples over $\\mathbb Z$. In this paper, we use the theory of ternary quadratic forms to show that 36 concrete tuples $(a,b,c,d,e,f)$ in Sun's list of candidates are indeed universal over $\\mathbb Z$. For example, we prove the universality of $(16,4,2,0,1,1)$ over $\\mathbb Z$ which is related to the famous form $x^2+y^2+32z^2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-19T17:59:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11E20",
      "11D85"
    ],
    "keywords": [
      "universal quadratic sums",
      "ternary quadratic forms",
      "suns list",
      "candidates",
      "universal tuples"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}