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

FINDING: Fermat's Last Theorem (FLT) — no integer solutions for \(x^n + y^n = z^n\), \(n>2\) — proven via modularity (elliptic curves ↔ modular forms), with a novel polynomial-root approach proposed for elementary proof. | MATH: Core equation: \(x^n + y^n = z^n\) (no nonzero integers for \(n>2\)). Wiles's proof: every semistable elliptic curve \(E: y^2 = x^3 + ax + b\) is modular (Taniyama–Shimura–Weil). Key invariant: conductor \(N_E\), discriminant \(\Delta_E\), \(j\)-invariant. Polynomial approach (arXiv:1105.0669): associate \(P_n(x) = (x^n + y^n - z^n)\) treated as polynomial in one variable; root analysis yields constraints on \(n\). | CONNECTION: **Direct geometric link**: The modularity theorem connects elliptic curves to **modular forms** — objects with **SL(2,Z) symmetry** (lattice transformations). The \(j\)-invariant has **Fourier expansion** \(j(q) = q^{-1} + 744 + 196884q + ...\) — the coefficients relate to **Monster group** (moonshine), a **crystallographic-like symmetr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com