{
  "id": "1312.7548",
  "version": "v2",
  "published": "2013-12-29T15:18:41.000Z",
  "updated": "2014-01-02T18:05:42.000Z",
  "title": "Proof of two divisibility properties of binomial coefficients conjectured by Z.-W. Sun",
  "authors": [
    "Victor J. W. Guo"
  ],
  "comment": "13 pages, add many new results",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "For all positive integers n, we prove the following divisibility properties: $$(2n+3){2n\\choose n} | 3{6n\\choose 3n}{3n\\choose n}, and (10n+3){3n\\choose n} | 21{15n\\choose 5n} {5n\\choose n}.$$ This confirms two recent conjectures of Z.-W. Sun. Some similar divisibility properties are given. Moreover, we show that, for all positive integers m and n, the product $am{am+bm-1\\choose am}{an+bn\\choose an}$ is divisible by m+n. In fact, the latter result can be generalized to the q-binomial coefficients and q-integers case, which generalizes the positivity of q-Catalan numbers. We also propose several related conjectures.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2014-01-02T18:05:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "05A10",
      "05A30"
    ],
    "keywords": [
      "binomial coefficients",
      "positive integers",
      "similar divisibility properties",
      "q-catalan numbers",
      "conjectures"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.7548G"
    }
  }
}