{
  "id": "1701.05868",
  "version": "v1",
  "published": "2017-01-20T17:34:21.000Z",
  "updated": "2017-01-20T17:34:21.000Z",
  "title": "Restricted sums of four squares",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "17 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two. For example, we show that any positive integer can be written as $x^2+y^2+z^2+w^2\\ (x,y,z,w\\in\\mathbb Z)$ with $x+y+z+w=2^{\\lfloor(\\mbox{ord}_2(n)+1)/2\\rfloor}$, and that each positive integer can be written as $x^2+y^2+z^2+w^2\\ (x,y,z,w\\in\\mathbb Z)$ with $x+y+z$ (or $x+y+2z$, or $x+2y+2z$) a power of four (including 1). We also pose some open conjectures; for example, we conjecture that any positive integer can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w$ nonnegative integers such that $x+2y-2z$ is a power of four, and that any positive integer can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w$ integers such that $x+2y+4z+8w\\in\\{8^k:\\ k=0,1,2,\\ldots\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-01-20T17:34:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11D85",
      "11E20",
      "11P05"
    ],
    "keywords": [
      "restricted sums",
      "positive integer",
      "refine lagranges four-square theorem",
      "open conjectures",
      "restrictions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}