{
  "id": "1907.12981",
  "version": "v1",
  "published": "2019-07-30T14:28:49.000Z",
  "updated": "2019-07-30T14:28:49.000Z",
  "title": "On two conjectures involving quadratic residues",
  "authors": [
    "Fedor Petrov",
    "Zhi-Wei Sun"
  ],
  "comment": "7 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime with $p\\equiv 1\\pmod 4$. In this paper we confirm two conjectures of Sun involving quadratic residues modulo $p$. For example, we show that for any integer $a\\not\\equiv0\\pmod p$ we have \\begin{align*}&(-1)^{|\\{1\\le k<p/4:\\ (\\frac kp)=-1\\}|}\\prod_{1\\le j<k\\le (p-1)/2}(e^{2\\pi iaj^2/p}+e^{2\\pi iak^2/p}) \\\\=&\\begin{cases}1&\\text{if}\\ p\\equiv1\\pmod 8,\\\\\\left(\\frac ap\\right)\\varepsilon_p^{-(\\frac ap)h(p)}&\\text{if}\\ p\\equiv5\\pmod8,\\end{cases} \\end{align*} where $(\\frac{\\cdot}p)$ is the Legendre symbol, and $\\varepsilon_p$ and $h(p)$ are the fundamental unit and the class number of the real quadratic field $\\mathbb Q(\\sqrt p)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-30T14:28:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A15",
      "11R11",
      "05A05",
      "11R32",
      "33B10"
    ],
    "keywords": [
      "conjectures",
      "quadratic residues modulo",
      "real quadratic field",
      "odd prime",
      "legendre symbol"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}