{
  "abstract": "For a prime $p\\ge7$, let $S(C_p\\times C_p)$ be the set of integer group determinants and $e(p)$ the least exponent for which $p^{e(p)}\\mathbf Z\\subseteq S(C_p\\times C_p)$. Let $r_p$ be the least integer $r\\ge1$ satisfying $(2r+1)i>r(p-1)$ for every irregular pair $(p,2i)$ with $r\\mid i$. We prove the unconditional inclusion $p^{p+3+2r_p}\\mathbf Z\\subseteq S(C_p\\times C_p)$. Equality $e(p)=p+3+2r_p$ holds when Vandiver's conjecture holds for $p$ and $r_p\\le(p-5)/2$, hence for every prime $7\\le p<2^{31}$. The first exception to $e(p)=p+5$ is $p=103$, where $e(103)=116$. A polynomial obstruction from Bernoulli numbers supplies the lower bound, and circular units give the construction. For regular primes, we characterize the entire determinant set through explicit affine systems over $\\mathbf F_p$ and principality conditions in $\\mathbf Z[ζ_p]$. For regular $p\\ge11$, prime cofactors cannot occur at valuation $p+3$; at valuation $p+4$ they have natural density $(p+1)/(hp^{(p-3)/2})$, where $h$ is the class number of $\\mathbf Q(ζ_p)$. We also show that $p^{p+5}$ is the least absolute value of a nonzero $p$-divisible determinant for these regular primes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15262",
  "author_total": 1,
  "authors": [
    {
      "name": "Chatchawan Panraksa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/33sqp2mc3vyww.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15262",
  "fulltext_url": null,
  "id": "33sqp2mc3vyww",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15262",
  "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|Analytic Number Theory Research"
    ]
  ],
  "openalex": "W7221803074",
  "pdf_url": "https://arxiv.org/pdf/2610.15262",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-11",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/33sqp2mc3vyww.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15262",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15262"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Bernoulli numbers and the integer group determinants of $C_p \\times C_p$",
  "title_text": "Bernoulli numbers and the integer group determinants of C_p × C_p",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/33sqp2mc3vyww",
  "venue": "arXiv",
  "versions": []
}