{
  "id": "0804.3750",
  "version": "v5",
  "published": "2008-04-23T15:56:45.000Z",
  "updated": "2009-02-07T09:43:28.000Z",
  "title": "Mixed sums of squares and triangular numbers (III)",
  "authors": [
    "Byeong-Kweon Oh",
    "Zhi-Wei Sun"
  ],
  "journal": "J. Number Theory 129(2009), no.4, 964-969",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper we confirm a conjecture of Sun which states that each positive integer is a sum of a square, an odd square and a triangular number. Given any positive integer m, we show that p=2m+1 is a prime congruent to 3 modulo 4 if and only if T_m=m(m+1)/2 cannot be expressed as a sum of two odd squares and a triangular number, i.e., p^2=x^2+8(y^2+z^2) for no odd integers x,y,z. We also show that a positive integer cannot be written as a sum of an odd square and two triangular numbers if and only if it is of the form 2T_m (m>0) with 2m+1 having no prime divisor congruent to 3 modulo 4.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2009-02-07T09:43:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "05A05",
      "11D85",
      "11P99",
      "11Y11"
    ],
    "keywords": [
      "triangular number",
      "mixed sums",
      "odd square",
      "positive integer",
      "prime divisor congruent"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0804.3750O"
    }
  }
}