Fermat's Last Theorem: Modularity Proof and Elementary Polynomial Reformulation — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — no integer solutions for \\(a^n + b^n = c^n\\) when \\(n > 2\\) — proven via the modularity (Taniyama-Shimura) of semistable elliptic curves; a polynomial-root reformulation offers an elementary alternative. MATH: - Core equation: \\(a^n + b^n = c^n\\), \\(n \\in \\mathbb{Z}^+\\), \\(n > 2\\) → no nonzero integer triples. - Wiles's proof: For a hypothetical solution, construct Frey curve \\(E: y^2 = x(x - a^n)(x + b^n)\\); its discriminant \\(\\Delta = (abc)^{2n}/16\\) has conductor \\(N\\) with specific prime divisibility. Ribet's level-lowering (epsilon conjecture) forces \\(E\\) to be modular of level 2, impossible — contradiction. - Modularity: \\(\\rho_{E,p} \\cong \\rho_{f,p}\\) for a weight-2 cusp form \\(f\\) of level \\(N\\); Wiles proved the full modularity for semistable curves via deformation rings and the \\(R = \\mathbb{T}\\) theorem. - Polynomial approach (arXiv:1105.0669v5): Associate to \\(a^n + b^n = c^n\\) the polynomial \\(P(x) = x^n - (a^n + b^n)\\); ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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Publication Details

Journal
Zenodo (CERN European Organization for Nuclear Research)
Published
2026-09-21
DOI
https://doi.org/10.5281/zenodo.22873702
Primary Topic
Cryptography and Residue Arithmetic
Type
preprint
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preprint

Fermat's Last Theorem: Modularity Proof and Elementary Polynomial Reformulation — E8 Intelligence Research

Andrew Stewart Caldin
Zenodo (CERN European Organization for Nuclear Research)
Cryptography and Residue Arithmetic
preprint

Fermat's Last Theorem: Modularity Proof and Elementary Polynomial Reformulation — E8 Intelligence Research

Andrew Stewart Caldin
preprint en

Abstract

FINDING: Fermat's Last Theorem (FLT) — no integer solutions for \(a^n + b^n = c^n\) when \(n > 2\) — proven via the modularity (Taniyama-Shimura) of semistable elliptic curves; a polynomial-root reformulation offers an elementary alternative. MATH: - Core equation: \(a^n + b^n = c^n\), \(n \in \mathbb{Z}^+\), \(n > 2\) → no nonzero integer triples. - Wiles's proof: For a hypothetical solution, construct Frey curve \(E: y^2 = x(x - a^n)(x + b^n)\); its discriminant \(\Delta = (abc)^{2n}/16\) has conductor \(N\) with specific prime divisibility. Ribet's level-lowering (epsilon conjecture) forces \(E\) to be modular of level 2, impossible — contradiction. - Modularity: \(\rho_{E,p} \cong \rho_{f,p}\) for a weight-2 cusp form \(f\) of level \(N\); Wiles proved the full modularity for semistable curves via deformation rings and the \(R = \mathbb{T}\) theorem. - Polynomial approach (arXiv:1105.0669v5): Associate to \(a^n + b^n = c^n\) the polynomial \(P(x) = x^n - (a^n + b^n)\); ro Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

Zenodo (CERN European Organization for Nuclear Research)
Reduced inequalities, Peace, Justice and strong institutions
Cryptography and Residue Arithmetic
DOIOpen Access PDFOpenAlex
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Fermat's Last Theorem: Modularity Proof and Elementary Polynomial Reformulation — E8 Intelligence Research — Andrew Stewart Caldin · Zenodo (CERN European Organization for Nuclear Research) (2026) | TGRS Research Map | TGRS