{
  "id": "1012.3919",
  "version": "v1",
  "published": "2010-12-17T16:38:00.000Z",
  "updated": "2010-12-17T16:38:00.000Z",
  "title": "Constructing $x^2$ for primes $p=ax^2+by^2$",
  "authors": [
    "Zhi-Hong Sun"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=ax^2+by^2$ for some integers $x$ and $y$. Let $\\lambda(a,b;n)$ be given by $q\\prod_{k=1}^\\infty (1-q^{ak})^3(1-q^{bk})^3 = \\sum_{n=1}^\\infty \\lambda(a,b;n)q^n$. In the paper, using Jacobi's identity $\\prod_{n=1}^\\infty (1-q^n)^3 = \\sum_{k=0}^\\infty (-1)^k(2k+1)q^{\\frac{k(k+1)}2}$ we construct $x^2$ in terms of $\\lambda(a,b;n)$. For example, if $2\\nmid ab$ and $p\\nmid ab(ab+1)$, then $(-1)^{\\frac{a+b}2x+\\frac{b+1}2}(4ax^2-2p) = \\lambda(a,b;((ab+1)p-a-b)/8+1)$. We also give formulas for $\\lambda(1,3;n+1),\\lambda(1,7;2n+1)$, $\\lambda(3,5;2n+1)$ and $\\lambda(1,15;4n+1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-12-17T16:38:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E16",
      "11E25"
    ],
    "keywords": [
      "odd prime",
      "jacobis identity",
      "positive integers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1012.3919S"
    }
  }
}