{
  "id": "2210.10575",
  "version": "v1",
  "published": "2022-10-19T14:15:40.000Z",
  "updated": "2022-10-19T14:15:40.000Z",
  "title": "Polynomial \\(D(4)\\)-quadruples over Gaussian Integers",
  "authors": [
    "Marija Bliznac Trebješanin",
    "Sanda Bujačić Babić"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "A set $\\{a, b, c, d\\}$ of four non-zero distinct polynomials in $\\mathbb{Z}[i][X]$ is said to be a Diophantine $D(4)$-quadruple if the product of any two of its distinct elements increased by 4 is a square of some polynomial in $\\mathbb{Z}[i][X]$. In this paper we prove that every $D(4)$-quadruple in $\\mathbb{Z}[i][X]$ is regular, or equivalently that the equation $$(a+b-c-d)^2=(ab+4)(cd+4)$$ holds for every $D(4)$-quadruple in $\\mathbb{Z}[i][X]$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-19T14:15:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D09",
      "11D45"
    ],
    "keywords": [
      "gaussian integers",
      "non-zero distinct polynomials",
      "distinct elements",
      "diophantine"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}