{
  "abstract": "For every quadratic number field $K$, we prove a uniform $3$-unknown Diophantine definition of integer tuples in $\\mathcal{O}_K$, allowing finitely many polynomial nonvanishing conditions. This yields an effective $+3$ transfer principle and a $13$-unknown representation of every recursively enumerable integer relation. Consequently, there exists an absolute degree bound $D_0\\geq1$ such that for every quadratic number field $K$, there is no algorithm that, given \\[ P(Y_1,\\ldots,Y_{13})\\in\\mathbb{Z}[Y_1,\\ldots,Y_{13}],\\quad \\mathrm{deg}\\ P\\leq D_0, \\] decides whether $P=0$ has a solution in $\\mathcal{O}_K^{13}$. The arithmetic input is a fourth-order Pell--Lucas congruence. It is a specialization of the norm-one Lucas multiplication formula, which yields exact valuations for the deviation of a Lucas quotient from its linear term, together with deviation criteria for Lucas--Wieferich and Wall--Sun--Sun primes. We also establish local surjectivity and $\\ell$-adic density for second-order correction terms for norm-one Lucas sequences.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30389",
  "author_total": 1,
  "authors": [
    {
      "name": "Geng-Rui Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/35mb16znp1av7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30389",
  "fulltext_url": null,
  "id": "35mb16znp1av7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30389",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.LO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204963779",
  "pdf_url": "https://arxiv.org/pdf/2608.30389",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/35mb16znp1av7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2608.30389",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30389"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.LO",
      "subfield": "logic-foundations"
    }
  ],
  "title": "$13$ unknowns over quadratic integer rings and Lucas congruences",
  "title_text": "$13$ unknowns over quadratic integer rings and Lucas congruences",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/35mb16znp1av7",
  "venue": "arXiv",
  "versions": []
}