{
  "id": "2008.10718",
  "version": "v1",
  "published": "2020-08-24T21:48:25.000Z",
  "updated": "2020-08-24T21:48:25.000Z",
  "title": "Four conjectures by Zhi-Hong Sun",
  "authors": [
    "Constantin N. Beli"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We prove some results conjectured by Zhi-Hong Sun regarding the value $\\mod p$ of $\\varepsilon_d^{\\frac{p-1}4}$, where $\\varepsilon_d$ is a unit of norm $-1$ in some fields $\\mathbb Q(\\sqrt d)$, with $\\left(\\frac{-1}p\\right) =\\left(\\frac dp\\right) =1$. The answer is given in terms of how $p$ writes as $p=f(x,y)=u^2+v^2$, with $x,y,u,v\\in\\mathbb Z$, where $f$ is a certain quadratic form of determinant $-4d$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-24T21:48:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B39",
      "11R37"
    ],
    "keywords": [
      "conjectures",
      "quadratic form",
      "zhi-hong sun regarding",
      "determinant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}