{
  "id": "2107.08984",
  "version": "v1",
  "published": "2021-07-19T15:58:31.000Z",
  "updated": "2021-07-19T15:58:31.000Z",
  "title": "A new theorem on quadratic residues modulo primes",
  "authors": [
    "Qing-Hu Hou",
    "Hao Pan",
    "Zhi-Wei Sun"
  ],
  "comment": "6 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p>3$ be a prime, and let $(\\frac{\\cdot}p)$ be the Legendre symbol. Let $b\\in\\mathbb Z$ and $\\varepsilon\\in\\{\\pm 1\\}$. We mainly prove that $$\\left|\\left\\{N_p(a,b):\\ 1<a<p\\ \\text{and}\\ \\left(\\frac ap\\right)=\\varepsilon\\right\\}\\right|=\\frac{3-(\\frac{-1}p)}2,$$ where $N_p(a,b)$ is the number of positive integers $x<p/2$ with $\\{x^2+b\\}_p>\\{ax^2+b\\}_p$, and $\\{m\\}_p$ with $m\\in\\mathbb{Z}$ is the least nonnegative residue of $m$ modulo $p$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-19T15:58:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A15",
      "11A07"
    ],
    "keywords": [
      "quadratic residues modulo primes",
      "legendre symbol",
      "positive integers",
      "nonnegative residue"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}