{
  "abstract": "We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in $\\mathbb{Z}[\\sqrt{-3}]$. A pigeonhole argument of Thue represents the relevant primes by $X^2+3Y^2$, and explicit division in $\\mathbb{Z}[\\sqrt{-3}]$ lifts primitive odd-power norms to powers in that order. For cubes, the resulting coordinate identities transform $2K(K^2+3M^2)=C^3$ into another equation of the same signed form with a strictly smaller non-zero integer height. The construction gives $|C'|\\le |C|/39$. Half-sums and half-differences of the two odd entries place the sum and difference parity configurations in this single descent.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.28566",
  "author_total": 1,
  "authors": [
    {
      "name": "R. Laniewski"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/26espbypvqz0r.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.28566",
  "fulltext_url": null,
  "id": "26espbypvqz0r",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.28566",
  "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.GM"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214373367",
  "pdf_url": "https://arxiv.org/pdf/2609.28566",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/26espbypvqz0r.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:15Z",
      "native_id": "2609.28566",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.28566"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2608|Algebraic Geometry and Number Theory",
      "subfield": "geometry-topology"
    }
  ],
  "title": "Primitive norm extraction and a signed descent for the cubic Fermat equation",
  "title_text": "Primitive norm extraction and a signed descent for the cubic Fermat equation",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:15Z",
  "url": "https://research.draconitic.com/paper/26espbypvqz0r",
  "venue": "arXiv",
  "versions": []
}