{
  "abstract": "For $d\\ge2$ the $q$-deformed modular group specialized at a primitive $d$-th root of unity is the triangle group of type $(2,3,d)$. Using this we determine the fractions $r/s$ for which the $d$-th cyclotomic polynomial divides the denominator $S_{r/s}(q)$ of the $q$-deformed rational number $[r/s]_q$. They form the orbit of $\\infty$ under the normal closure of the translation $z\\mapsto z+d$ in $\\mathrm{PSL}(2,\\mathbb{Z})$. This proves a conjecture of Byakuno, Ren and Yanagawa, and shows that the congruences $s\\equiv0$ and $r\\equiv\\pm1$ modulo $d$ characterize the divisibility exactly for $d\\le5$. For $a\\in\\{2,3,4,6\\}$ and $n>5a^2$ prime to $a$ we show that $S_{a/n}(q)$ is irreducible up to cyclotomic factors. Together with a computer check for small primes, this confirms a conjecture of Kogiso, Ren, Wakui, Yanagawa and the author for every prime $p$ and every $r$ prime to $p$ with $r\\equiv\\pm a$ or $ar\\equiv\\pm1\\pmod p$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.33703",
  "author_total": 1,
  "authors": [
    {
      "name": "Kengo Miyamoto"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/axja9pd9wr9hn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.33703",
  "fulltext_url": null,
  "id": "axja9pd9wr9hn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.33703",
  "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.CO"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "arxiv",
      "math.QA"
    ],
    [
      "openalex",
      "2602|Advanced Mathematical Identities"
    ]
  ],
  "openalex": "W7214723801",
  "pdf_url": "https://arxiv.org/pdf/2609.33703",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/axja9pd9wr9hn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.33703",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.33703"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cyclotomic factors and irreducibility of the denominators of $q$-deformed rational numbers",
  "title_text": "Cyclotomic factors and irreducibility of the denominators of q-deformed rational numbers",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/axja9pd9wr9hn",
  "venue": "arXiv",
  "versions": []
}