{
  "abstract": "Let $(a,b,c,d) = \\left(2mn, 2mp, m^2 - n^2 - p^2, m^2 + n^2 + p^2\\right)$ be a primitive Pythagorean quadruple and let $S=\\langle a, b, c, d\\rangle$ be the numerical semigroup generated by $a,b,c,$ and $d.$ For convenience, we also let $Q = n^2 + p^2, δ= \\gcd(n,p),$ and $n = δn_0$ for some $n_0 \\in \\mathbb{Z}$. In this paper, we determine the Frobenius number of $S$ and derive an explicit formula in terms of $m$, $n$, and $p$ with the assumption that $m\\geq 2Q$ for $δ= 1$ and $m \\geq \\frac{2Q}δ - 1$ for the remaining cases. The proof is based on an explicit complete residue system modulo $2mn$, a normalization procedure for arbitrary semigroup elements, and a lift-orbit description of boundary representatives.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21397",
  "author_total": 2,
  "authors": [
    {
      "name": "WonTae Hwang"
    },
    {
      "name": "Kyunghwan Song"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/akn9nbx0hwfsp.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21397",
  "fulltext_url": null,
  "id": "akn9nbx0hwfsp",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21397",
  "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.NT"
    ],
    [
      "openalex",
      "2602|Commutative Algebra and Its Applications"
    ]
  ],
  "openalex": "W7213939613",
  "pdf_url": "https://arxiv.org/pdf/2609.21397",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/akn9nbx0hwfsp.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21397",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21397"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Frobenius Numbers Associated with Primitive Pythagorean Quadruples",
  "title_text": "Frobenius Numbers Associated with Primitive Pythagorean Quadruples",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/akn9nbx0hwfsp",
  "venue": "arXiv",
  "versions": []
}