{
  "id": "2010.05775",
  "version": "v1",
  "published": "2020-10-12T16:57:14.000Z",
  "updated": "2020-10-12T16:57:14.000Z",
  "title": "Positive squares written as nontrivial sums of four squares",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $m$ be any positive integer, and let $\\lambda\\in\\{2,3\\}$. We show that $m^2$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w$ nonnegative integers such that $x+2y+\\lambda z$ is a positive square. We also pose some open conjectures; for example, we conjecture that any positive odd integer can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\\in\\{0,1,2,\\ldots\\}$ and $x+2y+3z\\in\\{2^a:\\ a=1,2,3,\\ldots\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-12T16:57:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11D85",
      "11E20"
    ],
    "keywords": [
      "positive squares written",
      "nontrivial sums",
      "open conjectures",
      "positive odd integer",
      "nonnegative integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}