Fermat's Last Theorem: Modularity Proof and a Novel Polynomial-Root Approach — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — no integer solutions for \\(x^n + y^n = z^n\\) for \\(n>2\\) — proven via the Taniyama–Shimura (modularity) conjecture; a novel polynomial-root approach is proposed in arXiv:1105.0669v5. MATH: Core equation: \\(x^n + y^n = z^n\\), \\(n \\in \\mathbb{Z}^+\\), \\(n>2\\). Wiles' proof: every semistable elliptic curve over \\(\\mathbb{Q}\\) is modular — i.e., its \\(L\\)-function equals that of a weight-2 modular form. Key objects: Frey curve \\(E: y^2 = x(x-a^n)(x+b^n)\\) (with \\(a^n+b^n=c^n\\)), Galois representations, Ribet's level-lowering theorem, and the modularity theorem. The arXiv paper associates a polynomial \\(P_n(x)\\) of degree \\(n\\) to each FLT equation; roots of \\(P_n\\) are studied to test FLT validity — a method claimed to be within Fermat's reach. CONNECTION: The Frey curve's discriminant \\(\\Delta = (abc)^{2n}\\) and \\(j\\)-invariant relate to the ratio \\((a^n b^n c^n)\\) — but no direct golden-ratio or base-60 link. However, the modular forms involved l Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com