{
  "abstract": "For integers $m\\geq 2$ and $m\\mid N$, let $G(N;m)=\\gcd\\{\\binom{N}{k}:0<k<N,\\ m\\mid k\\}$. We prove a complete $p$-adic valuation formula for $G(N;m)$ at primes $p\\equiv -1\\pmod m$, under the hypotheses $m\\geq 3$, $m\\mid N$, and $m<N$. Writing $N=\\sum_i d_i p^i$, put $A=\\sum_{i\\text{ even}}d_i$ and $B=\\sum_{i\\text{ odd}}d_i$. Then $v_p(G(N;m))$ is $2$ in the exceptional mixed case $(A,B)=(1,1)$ with $p=m-1$, is $1$ in the mixed case $(A,B)=(1,1)$ with $m<p$, is $1$ in the one-parity cases $(A,B)=(m,0)$ and $(A,B)=(0,m)$, and is $0$ otherwise. The proof uses Kummer's theorem to translate the problem into digitwise borrow counts and a minimal signed zero-sum classification. A source-level literature audit through 25 September 2026 located no equivalent prior theorem for the full minus-one classification: McTague's corrected same-residue extension covers the one-parity $p>m$ subcases, but not the mixed-parity branch or the $p=m-1$ regime. The theorem, its plus-one companion, a scaling reduction, and the $m=3,4,6$ specializations have been formalized in Lean 4 / Mathlib.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.37754",
  "author_total": 1,
  "authors": [
    {
      "name": "John Fairfax-Ball"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/fpw05x42r2dre.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.37754",
  "fulltext_url": null,
  "id": "fpw05x42r2dre",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.37754",
  "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": "W7214970570",
  "pdf_url": "https://arxiv.org/pdf/2609.37754",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/fpw05x42r2dre.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:35Z",
      "native_id": "2609.37754",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.37754"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Restricted Binomial GCDs at Primes Congruent to -1",
  "title_text": "Restricted Binomial GCDs at Primes Congruent to -1",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:35Z",
  "url": "https://research.draconitic.com/paper/fpw05x42r2dre",
  "venue": "arXiv",
  "versions": []
}