{
  "id": "2305.09267",
  "version": "v1",
  "published": "2023-05-16T08:20:18.000Z",
  "updated": "2023-05-16T08:20:18.000Z",
  "title": "On orders in quadratic number fields whose set of distances is peculiar",
  "authors": [
    "Andreas Reinhart"
  ],
  "categories": [
    "math.NT",
    "math.AC"
  ],
  "abstract": "Let $\\mathcal{O}$ be an order in an algebraic number field and suppose that the set of distances $\\Delta(\\mathcal{O})$ of $\\mathcal{O}$ is nonempty (equivalently, $\\mathcal{O}$ is not half-factorial). If $\\mathcal{O}$ is seminormal (in particular, if $\\mathcal{O}$ is a principal order), then $\\min\\Delta(\\mathcal{O})=1$. So far, only a few examples of orders were found with $\\min\\Delta(\\mathcal{O})>1$. We say that $\\Delta(\\mathcal{O})$ is peculiar if $\\min\\Delta(\\mathcal{O})>1$. In the present paper, we establish algebraic characterizations of orders $\\mathcal{O}$ in real quadratic number fields with $\\min\\Delta(\\mathcal{O})>1$. We also provide a classification of the real quadratic number fields that possess an order whose set of distances is peculiar. As a consequence thereof, we revisit certain squarefree integers (cf. OEIS A135735) that were studied by A.J. Stephens and H.C. Williams.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-16T08:20:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R11",
      "11R27",
      "13A15",
      "13F15",
      "20M12",
      "20M13"
    ],
    "keywords": [
      "real quadratic number fields",
      "algebraic number field",
      "establish algebraic characterizations",
      "principal order",
      "consequence thereof"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}