{
  "id": "1505.03679",
  "version": "v1",
  "published": "2015-05-14T16:00:43.000Z",
  "updated": "2015-05-14T16:00:43.000Z",
  "title": "On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "8 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be any of the following six triples: $$(1,2,3),\\ (1,2,4),\\ (1,2,5),\\ (2,2,4),\\ (2,2,5),\\ (2,3,3),\\ (2,3,4),$$ and conjecture that any triple $(a,b,c)$ among $$(2,2,6),\\ (2,3,5),\\ (2,3,7),\\ (2,3,8),\\ (2,3,9),\\ (2,3,10)$$ also has that property. For integers $0\\le b\\le c\\le d\\le a$ with $a>2$, we prove that any nonnegative integer can be represented as $x(ax+b)+y(ay+c)+z(az+d)$ with $x,y,z$ integers, if and only if the quadruple $(a,b,c,d)$ is among $$(3,0,1,2),\\ (3,1,1,2),\\ (3,1,2,2),\\ (3,1,2,3),\\ (4,1,2,3).$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-05-14T16:00:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11D85",
      "11E20"
    ],
    "keywords": [
      "nonnegative integer",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150503679S"
    }
  }
}