Fermat's Last Theorem: A Novel Polynomial-Root Reinterpretation via Elliptic Curves — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — no integer solutions for \\(a^n + b^n = c^n\\), \\(n>2\\) — proven via elliptic curves (Taniyama-Shimura/Modularity), with a novel polynomial-root reinterpretation in the arXiv paper. | MATH: Core equation: \\(x^n + y^n = z^n\\) (n>2, no nonzero integers). Wiles' proof: every semistable elliptic curve \\(E: y^2 = x^3 + ax + b\\) is modular (a cusp form of weight 2). Frey curve: \\(y^2 = x(x - a^p)(x + b^p)\\) — if FLT counterexample existed, this curve is non-modular, contradiction. The arXiv paper (1105.0669v5) associates polynomial \\(P_n(x) = x^n + y^n - z^n\\) (treating y,z as parameters) and studies roots to infer FLT validity — a finite-field or algebraic-geometry approach. | CONNECTION: The Frey curve's discriminant \\(\\Delta = (abc)^{2p}\\) and \\(j\\)-invariant contain powers of 2 and 3 — linking to modular forms whose Fourier coefficients live in \\(\\mathbb{Q}\\) with ramification at primes 2,3. The modularity theorem itself is a statement of symmetry: el Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com