{
  "id": "1504.01608",
  "version": "v3",
  "published": "2015-04-06T15:56:21.000Z",
  "updated": "2015-04-13T17:14:25.000Z",
  "title": "Natural numbers represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor+\\lfloor z^2/c\\rfloor$",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "20 pages. Expanded version with new results and conjectures added",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\\in\\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor +\\lfloor z^2/c\\rfloor$ with $x,y,z\\in\\mathbb Z$ if $(a,b,c)\\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\\lfloor T_x/a\\rfloor+\\lfloor T_y/b\\rfloor+\\lfloor T_z/c\\rfloor$ with $x,y,z\\in\\mathbb Z$, where $T_x$ denotes the triangular number $x(x+1)/2$. We confirm this general conjecture in some special cases; in particular, we prove that $\\{x^2+y^2+\\lfloor z^2/m\\rfloor:\\ x,y,z\\in\\mathbb Z\\}=\\{0,1,2,...\\}$ for all $m\\in\\{2,3,4,5,6,8,9,21\\}$. We also conjecture that for any real number $\\alpha\\in(0,1.5]$ with $\\alpha\\not=1$ each positive integer can be written as the sum of three elements of the set $\\{x^2+\\lfloor \\alpha x\\rfloor:\\ x\\in\\mathbb Z\\}$ one of which is odd.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-04-06T15:56:21.000Z",
      "abstract": "Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\\in\\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor +\\lfloor z^2/c\\rfloor$ with $x,y,z\\in\\mathbb Z$ if $(a,b,c)\\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\\lfloor T_x/a\\rfloor+\\lfloor T_y/b\\rfloor+\\lfloor T_z/c\\rfloor$ with $x,y,z\\in\\mathbb Z$, where $T_x$ denotes the triangular number $x(x+1)/2$. We confirm this general conjecture in some special cases; in particular, we prove that $\\{x^2+y^2+\\lfloor z^2/m\\rfloor:\\ x,y,z\\in\\mathbb Z\\}=\\{0,1,2,...\\}$ for all $m\\in\\{2,3,4,5,6,8,9,21\\}$. We also conjecture that for any real number $\\alpha\\in(0,1.5)$ with $\\alpha\\not=1$ each positive integer can be written as the sum of three elements of the set $\\{x^2+\\lfloor \\alpha x\\rfloor:\\ x\\in\\mathbb Z\\}$ one of which is odd.",
      "comment": "11 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-04-13T17:14:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11B75",
      "11D85",
      "11E20",
      "11P32"
    ],
    "keywords": [
      "natural numbers",
      "positive integer",
      "special cases",
      "real number",
      "general conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150401608S"
    }
  }
}