{
  "id": "2408.07034",
  "version": "v1",
  "published": "2024-08-13T16:59:15.000Z",
  "updated": "2024-08-13T16:59:15.000Z",
  "title": "On a determinant involving linear combinations of Legendre symbols",
  "authors": [
    "Keqin Liu",
    "Zhi-Wei Sun",
    "Li-Yuan Wang"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "In this paper, we study a conjecture of the second author on determinants involving Legendre symbols. For example, we prove that for any prime $p\\equiv3\\pmod4$, where $(\\frac{\\cdot}p)$ is the Legendre symbol: $$\\det\\left[x+\\left(\\frac{i-j}p\\right)+\\left(\\frac ip\\right)y+\\left(\\frac jp\\right)z+\\left(\\frac{ij}p\\right)w\\right]_{0\\le i,j\\le(p-3)/2}=x$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-13T16:59:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11A15",
      "11C20",
      "15A15"
    ],
    "keywords": [
      "legendre symbol",
      "linear combinations",
      "determinant"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}