Fermat's Last Theorem: Elliptic Curves, Modular Forms, and an Elementary Claim — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — no integer solutions for \(x^n + y^n = z^n\), \(n>2\) — proven via elliptic curves and modular forms; one novel polynomial-root approach claims elementary insight. | MATH: FLT: \(x^n + y^n = z^n\) has no nonzero integer solutions for \(n>2\). Wiles–Taylor proof: every semistable elliptic curve over \(Q\) is modular (Taniyama–Shimura–Weil). Key objects: Frey curve \(E: y^2 = x(x-a^n)(x+b^n)\), Galois representations, modular forms of weight 2, level \(N\). The arXiv paper (1105.0669v5) constructs a polynomial \(P_n(x)\) whose roots encode FLT failure; claims root analysis yields contradiction for \(n≥ 3\). | CONNECTION: The Frey curve's discriminant \(Δ = (abc)²ⁿ\) and conductor \(N\) involve prime factors — but no direct golden ratio or base-60 link. However, modular forms live on the upper half-plane with symmetry group \(SL(2,Z)\) — a lattice symmetry (root system \(A_1\) in disguise). The elliptic curve's \(j\)-invar Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com