{
  "id": "2008.10502",
  "version": "v1",
  "published": "2020-08-24T15:08:29.000Z",
  "updated": "2020-08-24T15:08:29.000Z",
  "title": "Legendre Symbol of $\\prod f(i,j) $ over $ 0<i<j<p/2, \\ p\\nmid f(i,j) $",
  "authors": [
    "Chao Huang"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p>3$ be a prime. We investigate Legendre symbol of $\\displaystyle \\prod_{0<i<j<p/2 \\atop p\\nmid f(i,j) } f(i,j) \\ $, where $i,j\\in \\Bbb Z, f(i,j)$ is a linear or quadratic form with integer coefficients. When $f=ai^2+bij+cj^2$ and $p\\nmid c(a+b+c)$ , we prove that to evaluate the product is equivalent to determine $ \\displaystyle \\sum_{y=1}^{p-1} \\bigg(\\frac{y(y+1)(y+k)}{p}\\bigg) \\pmod{16}$ , where $4c(a+b+c)k \\equiv (4ac-b^2)\\pmod{p}.$ Parallel results are given for $\\displaystyle \\prod_{i,j=1 \\atop p\\nmid f(i,j) }^{(p-1)/2} \\bigg(\\frac{ f(i,j) }{p}\\bigg).$ Then we show that $ \\displaystyle \\sum_{y=1}^{p-1} \\bigg(\\frac{y(y+1)(y+k)}{p}\\bigg) \\pmod{16}$ can be evaluated explicitly when k=2,4,5,9,10 or k is a square. And for several classes of f(i,j) these two kinds of products can be evaluated explicitly. Finally when f is a linear form we give unified identities for these products. Thus we prove these kind of problems raised in Sun's paper.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-24T15:08:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A07",
      "11A15",
      "11T24"
    ],
    "keywords": [
      "legendre symbol",
      "integer coefficients",
      "quadratic form",
      "parallel results",
      "linear form"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}