{
  "abstract": "Let $K$ be a number field and $A,B,C\\in\\mathcal{O}_K\\setminus\\{0\\}$. Put $S_K=\\{\\mathfrak{P}:\\mathfrak{P}\\mid2\\}$ and $S_K'=\\{\\mathfrak{q}:\\mathfrak{q}\\mid2ABC\\}$. We establish two criteria for the non-existence of non-trivial solutions of $Ax^p+By^p=Cz^p$ over $K$ for sufficiently large prime exponents $p$. Both criteria use a valuation bound on the $S_K'$-unit equation at a fixed $\\mathfrak{P}\\in S_K$. The first excludes all non-trivial solutions in $K^3$ when $v_{\\mathfrak{P}}(A)=v_{\\mathfrak{P}}(B)=m$ and $v_{\\mathfrak{P}}(C)=m+k$, with $k>4v_{\\mathfrak{P}}(2)$. The second, under an explicit nonvanishing condition on the coefficient valuations, excludes solutions $(a,b,c)\\in(K^\\times)^3$ whose $\\mathfrak{P}$-valuations are not all equal. In particular, it excludes non-trivial primitive integral solutions with $\\mathfrak{P}\\mid abc$. Neither criterion restricts the residue degree of $\\mathfrak{P}$. Both criteria hold unconditionally over every totally real field; over general number fields, we assume residual modularity and Eichler--Shimura-type conjectures. The first criterion covers a new coefficient range. In particular, under the corresponding $S_K$-unit bound, $x^p+y^p=2^rz^p$ has no non-trivial solutions in $K^3$ for each fixed $r\\geq5$ and sufficiently large prime $p$. This bound holds for a set of real quadratic fields of relative density $1$. For a family of imaginary quadratic fields of relative density $5/6$, we explicitly determine the $S_K$-unit solutions and verify the valuation bound for both criteria when $A,B,C\\in\\{\\pm2^r:r\\in\\mathbb{Z}_{\\geq0}\\}$, including the same $r\\geq5$ conclusion.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28117",
  "author_total": 1,
  "authors": [
    {
      "name": "Satyabrat Sahoo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/f9cyhaegmk9kf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28117",
  "fulltext_url": null,
  "id": "f9cyhaegmk9kf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28117",
  "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.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7204764336",
  "pdf_url": "https://arxiv.org/pdf/2608.28117",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/f9cyhaegmk9kf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:32Z",
      "native_id": "2608.28117",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28117"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Generalized Fermat equation over number fields",
  "title_text": "Generalized Fermat equation over number fields",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/f9cyhaegmk9kf",
  "venue": "arXiv",
  "versions": []
}