{
  "id": "1502.06454",
  "version": "v1",
  "published": "2015-02-23T14:58:35.000Z",
  "updated": "2015-02-23T14:58:35.000Z",
  "title": "Arithmetic Identities and Congruences for Partition Triples with 3-cores",
  "authors": [
    "Liuquan Wang"
  ],
  "comment": "14 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let ${{B}_{3}}(n)$ denote the number of partition triples of $n$ where each partition is 3-core. With the help of generating function manipulations, we find several infinite families of arithmetic identities and congruences for ${{B}_{3}}(n)$. Moreover, let $\\omega (n)$ denote the number of representations of a nonnegative integer $n$ in the form $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+3y_{1}^{2}+3y_{2}^{2}+3y_{3}^{2}$ with ${{x}_{1}},{{x}_{2}},{{x}_{3}},{{y}_{1}},{{y}_{2}},{{y}_{3}}\\in \\mathbb{Z}.$ We find three arithmetic relations between ${{B}_{3}}(n)$ and $\\omega (n)$, such as $\\omega (6n+5)=4{{B}_{3}}(6n+4).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-02-23T14:58:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A17",
      "11P83"
    ],
    "keywords": [
      "partition triples",
      "arithmetic identities",
      "congruences",
      "generating function manipulations",
      "arithmetic relations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}