{
  "id": "1103.4325",
  "version": "v11",
  "published": "2011-03-22T17:46:57.000Z",
  "updated": "2014-02-20T13:16:55.000Z",
  "title": "Conjectures and results on $x^2$ mod $p^2$ with $4p=x^2+dy^2$",
  "authors": [
    "Zhi-Wei Sun"
  ],
  "comment": "Provide final publication information",
  "journal": "in: Number Theory and Related Area (eds., Y. Ouyang, C. Xing, F. Xu and P. Zhang), Adv. Lect. Math. 27, Higher Education Press & International Press, Beijing-Boston, 2013, pp. 149-197",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "Given a squarefree positive integer $d$, we want to find integers (or rational numbers with denominators not divisible by large primes) $a_0,a_1,a_2,\\ldots$ such that for sufficiently large primes $p$ we have $\\sum_{k=0}^{p-1}a_k\\equiv x^2-2p$ (mod $p^2$) if $4p=x^2+dy^2$ (and $4\\nmid x$ if $d=1$), and $\\sum_{k=0}^{p-1}a_k\\equiv 0$ (mod $p^2$) if $(\\frac{-d}p)=-1$. In this paper we give a survey of conjectures and results on this topic and point out the connection between this problem and series for $1/\\pi$.",
  "revisions": [
    {
      "version": "v11",
      "updated": "2014-02-20T13:16:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11B65",
      "11A07",
      "05A10"
    ],
    "keywords": [
      "conjectures",
      "squarefree positive integer",
      "sufficiently large primes",
      "rational numbers",
      "denominators"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.4325S"
    }
  }
}