{
  "id": "2010.09200",
  "version": "v1",
  "published": "2020-10-19T04:08:09.000Z",
  "updated": "2020-10-19T04:08:09.000Z",
  "title": "There is no Diophantine $D(-1)$--quadruple",
  "authors": [
    "Nicolae Ciprian Bonciocat",
    "Mihai Cipu",
    "Maurice Mignotte"
  ],
  "comment": "29 pages, 2 figures",
  "categories": [
    "math.NT"
  ],
  "abstract": "A set of positive integers with the property that the product of any two of them is the successor of a perfect square is called Diophantine $D(-1)$--set. Such objects are usually studied via a system of generalized Pell equations naturally attached to the set under scrutiny. In this paper, an innovative technique is introduced in the study of Diophantine $D(-1)$--quadruples. The main novelty is the uncovering of a quadratic equation relating various parameters describing a hypothetical $D(-1)$--quadruple with integer entries. In combination with extensive computations, this idea leads to the confirmation of the conjecture according to which there is no Diophantine $D(-1)$--quadruple.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-10-19T04:08:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11D09",
      "11D45",
      "11B37",
      "11J68"
    ],
    "keywords": [
      "diophantine",
      "perfect square",
      "generalized pell equations",
      "main novelty",
      "quadratic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 29,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}