{
  "id": "0907.1725",
  "version": "v4",
  "published": "2009-07-10T05:09:16.000Z",
  "updated": "2012-07-04T15:12:54.000Z",
  "title": "On representation of an integer as a sum by X^2+Y^2+Z^2 and the modular equations of degree 3 and 5",
  "authors": [
    "Alexander Berkovich"
  ],
  "comment": "16 pages, To appear in the volume \"Quadratic and Higher Degree Forms\", in Developments in Math., Springer 2012",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "I discuss a variety of results involving s(n), the number of representations of n as a sum of three squares. One of my objectives is to reveal numerous interesting connections between the properties of this function and certain modular equations of degree 3 and 5. In particular, I show that s(25n)=(6-(-n|5))s(n)-5s(n/25) follows easily from the well known Ramanujan modular equation of degree 5. Moreover, I establish new relations between s(n) and h(n), g(n), the number of representations of $n$ by the ternary quadratic forms 2x^2+2y^2+2z^2-yz+zx+xy and x^2+y^2+3z^2+xy, respectively. I propose an interesting identity for s(p^2n)- p s(n) with p being an odd prime. This identity makes nontrivial use of the ternary quadratic forms with discriminants p^2, 16p^2.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-07-04T15:12:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E20",
      "11F37",
      "11B65",
      "05A30",
      "E05"
    ],
    "keywords": [
      "ternary quadratic forms",
      "representation",
      "ramanujan modular equation",
      "reveal numerous interesting connections",
      "odd prime"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.1725B"
    }
  }
}