{
  "abstract": "We give a deterministic polynomial-time primality test whose main verification is the congruence $(1+X)^n=1+X^n$ in $C_n[X]/(X^r-b(Y))$, where $C_n=(\\mathbb{Z}/n\\mathbb{Z})[Y]/(Φ_r(Y))$. Two identities in the coefficient ring normalize $b$ and ensure compatibility with the main congruence. A finite-field root count and a lattice bound show that an accepted input is a prime power; elementary preprocessing excludes nontrivial perfect powers. We construct $b$ deterministically using Jacobi sums and the uniform Dedekind-zeta zero-free theorem of Cureton and Thorner, which extends OpenAI's Hecke zero-free theorem. The total bit complexity is $L^{32+o(1)}$, where $L=\\lceil\\log_2 n\\rceil$, and verification after constructing $b$ costs $L^{6+o(1)}$. More generally, for a fixed real algebraic $1/2\\leθ<1$, a common zero-free boundary $\\Re s>θ$ for the relevant Dedekind zeta functions gives $L^{\\max\\{6,(1-θ)^{-1}\\}+o(1)}$. This bound reaches $L^{6+o(1)}$ at $θ=5/6$, and hence under GRH. We state a fixed-base modification of Agrawal's conjecture for odd integers passing the base-$2$ Fermat test, using the modulus $X^r-2$ with $r$ the least positive integer not dividing $n(n-1)$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.02280",
  "author_total": 1,
  "authors": [
    {
      "name": "Joseph M. Shunia"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1cd2yhky9jsfw.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.02280",
  "fulltext_url": null,
  "id": "1cd2yhky9jsfw",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.02280",
  "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",
      "cs.CC"
    ],
    [
      "arxiv",
      "math.NT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.02280",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1cd2yhky9jsfw.ris",
  "sources": [
    {
      "first_seen": "2026-11-03T12:21:10Z",
      "native_id": "2611.02280",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.02280"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence",
  "title_text": "Deterministic Polynomial-Time Primality Testing with a Single Binomial Congruence",
  "type": "preprint",
  "updated_at": "2026-11-05T15:20:22Z",
  "url": "https://research.draconitic.com/paper/1cd2yhky9jsfw",
  "venue": "arXiv",
  "versions": []
}