{
  "id": "0912.2671",
  "version": "v4",
  "published": "2009-12-14T19:26:25.000Z",
  "updated": "2009-12-20T04:51:41.000Z",
  "title": "Curious congruences for Fibonacci numbers",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "16 pages. Revise Conj. 4.2",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper we establish some sophisticated congruences involving central binomial coefficients and Fibonacci numbers. For example, we show that if $p\\not=2,5$ is a prime then $$\\sum_{k=0}^{p-1}F_{2k}\\binom{2k}{k}=(-1)^{[p/5]}(1-(p/5)) (mod p^2)$$ and $$\\sum_{k=0}^{p-1}F_{2k+1}\\binom{2k}k=(-1)^{[p/5]}(p/5) (mod p^2).$$ We also obtain similar results for some other second-order recurrences and raise several conjectures.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2009-12-20T04:51:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11B65",
      "05A10",
      "11A07"
    ],
    "keywords": [
      "fibonacci numbers",
      "curious congruences",
      "central binomial coefficients",
      "similar results",
      "second-order recurrences"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0912.2671S"
    }
  }
}