{
  "id": "2004.09155",
  "version": "v1",
  "published": "2020-04-20T09:32:04.000Z",
  "updated": "2020-04-20T09:32:04.000Z",
  "title": "A pair of congruences concerning sums of central binomial coefficients",
  "authors": [
    "Guo-Shuai Mao",
    "Roberto Tauraso"
  ],
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper, we prove the following congruences: for any prime $p\\equiv1\\pmod3$: $$\\sum_{k=1}^{\\lfloor\\frac{2p}3\\rfloor}\\binom{2k}{k}(-2)^k\\equiv0\\pmod{p^2}\\qquad \\text{ and }\\qquad \\sum_{k=0}^{\\lfloor\\frac{5p}6\\rfloor}\\frac{\\binom{2k}k}{(-32)^k}\\equiv\\left(\\frac2p\\right)\\pmod{p^2}$$ where $\\left(\\frac{\\cdot}{p}\\right)$ stands for the Legendre symbol.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-20T09:32:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "05A10",
      "11B65",
      "11G05",
      "33B15"
    ],
    "keywords": [
      "central binomial coefficients",
      "congruences concerning sums",
      "legendre symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}