{
  "id": "2007.06453",
  "version": "v1",
  "published": "2020-07-08T15:57:02.000Z",
  "updated": "2020-07-08T15:57:02.000Z",
  "title": "Proof of three conjectures on determinants related to quadratic residues",
  "authors": [
    "Darij Grinberg",
    "Zhi-Wei Sun",
    "Lilu Zhao"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper we confirm three conjectures of Z.-W. Sun on determinants. We first show that any odd integer $n>3$ divides the determiant $$\\left|(i^2+dj^2)\\left(\\frac{i^2+dj^2}n\\right)\\right|_{0\\le i,j\\le (n-1)/2},$$ where $d$ is any integer and $(\\frac{\\cdot}n)$ is the Jacobi symbol. Then we prove some divisibility results concerning $|(i+dj)^n|_{0\\le i,j\\le n-1}$ and $|(i^2+dj^2)^n|_{0\\le i,j\\le n-1}$, where $d\\not=0$ and $n>2$ are integers. Finally, for any odd prime $p$ and integers $c$ and $d$ with $p\\nmid cd$, we determine completely the Legendre symbol $(\\frac{S_c(d,p)}p)$, where $S_c(d,p):=|(\\frac{i^2+dj^2+c}p)|_{1\\le i,j\\le(p-1)/2}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-07-08T15:57:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11C20",
      "11A07",
      "11A15",
      "15A15"
    ],
    "keywords": [
      "quadratic residues",
      "determinants",
      "conjectures",
      "jacobi symbol",
      "odd integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}