{
  "abstract": "We introduce analogues of Diophantine $D(N)$-$m$-tuples in the noncommutative ring $M_2(\\mathbb Z)$ of $2\\times2$ integer matrices. Besides definitions based on the standard matrix product, we consider a symmetric version defined via the Jordan product $$A\\circ B=\\frac12(AB+BA).$$ Special attention is devoted to upper-triangular integer matrices $UT_2(\\mathbb Z)$, where squares admit a particularly simple description. Motivated by the classical connection between representations of $n$ as a difference of two squares and the existence of $D(n)$-quadruples in commutative rings, we investigate the existence of Jordan $D(N)$-quadruples in $UT_2(\\mathbb Z)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.29968",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrej Dujella"
    },
    {
      "name": "Zrinka Franušić"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/92z6g3s7k891c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.29968",
  "fulltext_url": null,
  "id": "92z6g3s7k891c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.29968",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2602|Rings, Modules, and Algebras"
    ]
  ],
  "openalex": "W7204890874",
  "pdf_url": "https://arxiv.org/pdf/2608.29968",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/92z6g3s7k891c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.29968",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.29968"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "$D(N)$-quadruples in upper-triangular $2\\times2$ integer matrices",
  "title_text": "D(N)-quadruples in upper-triangular 2×2 integer matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/92z6g3s7k891c",
  "venue": "arXiv",
  "versions": []
}