{
  "id": "2008.06675",
  "version": "v1",
  "published": "2020-08-15T08:20:58.000Z",
  "updated": "2020-08-15T08:20:58.000Z",
  "title": "A $p$-adic analogue of Chan and Verrill's formula for $1/π$",
  "authors": [
    "Ji-Cai Liu"
  ],
  "comment": "12 pages",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "We prove three supercongruences for sums of Almkvist-Zudilin numbers, which confirm some conjectures of Zudilin and Z.-H. Sun. A typical example is the Ramanujan-type supercongruence: \\begin{align*} \\sum_{k=0}^{p-1} \\frac{4k+1}{81^k}\\gamma_k \\equiv \\left(\\frac{-3}{p}\\right) p\\pmod{p^3}, \\end{align*} which is corresponding to Chan and Verrill's formula for $1/\\pi$: \\begin{align*} \\sum_{k=0}^\\infty \\frac{4k+1}{81^k}\\gamma_k = \\frac{3\\sqrt{3}}{2\\pi}. \\end{align*} Here $\\gamma_n$ are the Almkvist-Zudilin numbers.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-15T08:20:58.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11B65",
      "11Y55",
      "05A19"
    ],
    "keywords": [
      "verrills formula",
      "adic analogue",
      "almkvist-zudilin numbers",
      "ramanujan-type supercongruence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}