{
  "id": "1010.2489",
  "version": "v5",
  "published": "2010-10-12T19:57:46.000Z",
  "updated": "2014-08-07T11:56:44.000Z",
  "title": "Proof of three conjectures on congruences",
  "authors": [
    "Hao Pan",
    "Zhi-Wei Sun"
  ],
  "comment": "16 pages, final published version",
  "journal": "Sci. China Math. 57(2014), 2091-2102",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "In this paper we prove three conjectures on congruences involving central binomial coefficients or Lucas sequences. Let $p$ be an odd prime and let $a$ be a positive integer. We show that if $p\\equiv 1\\pmod{4}$ or $a>1$ then $$ \\sum_{k=0}^{\\lfloor\\frac34p^a\\rfloor}\\binom{-1/2}k\\equiv\\left(\\frac{2}{p^a}\\right)\\pmod{p^2}, $$ where $(-)$ denotes the Jacobi symbol. This confirms a conjecture of the second author. We also confirm a conjecture of R. Tauraso by showing that $$\\sum_{k=1}^{p-1}\\frac{L_k}{k^2}\\equiv0\\pmod{p}\\quad {\\rm provided}\\ \\ p>5,$$ where the Lucas numbers $L_0,L_1,L_2,\\ldots$ are defined by $L_0=2,\\ L_1=1$ and $L_{n+1}=L_n+L_{n-1}\\ (n=1,2,3,\\ldots)$. Our third theorem states that if $p\\not=5$ then we can determine $F_{p^a-(\\frac{p^a}5)}$ mod $p^3$ in the following way: $$\\sum_{k=0}^{p^a-1}(-1)^k\\binom{2k}k\\equiv\\left(\\frac{p^a}5\\right)\\left(1-2F_{p^a-(\\frac{p^a}5)}\\right)\\ \\pmod{p^3},$$ which appeared as a conjecture in a paper of Sun and Tauraso in 2010.",
  "revisions": [
    {
      "version": "v5",
      "updated": "2014-08-07T11:56:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B65",
      "11A07",
      "05A10",
      "11B39"
    ],
    "keywords": [
      "conjecture",
      "congruences",
      "central binomial coefficients",
      "third theorem states",
      "jacobi symbol"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.2489P"
    }
  }
}