Fermat's Last Theorem: A Polynomial-Roots Alternative to Wiles' Proof — 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 (Taniyama-Shimura); a polynomial-roots approach is proposed as an alternative. | MATH: Core equation: \(x^n + y^n = z^n\) (n>2, integers). Wiles' proof: every semistable elliptic curve \(E: y^2 = x^3 + ax + b\) is modular (a cusp form of weight 2, level N). The polynomial approach (arXiv:1105.0669) associates \(P(t) = (t^n - 1)\) or similar to FLT, analyzing roots to show no rational solutions for n≥3. Key constants: none new, but the Frey curve \(y^2 = x(x - a^n)(x + b^n)\) links FLT to modular forms. | CONNECTION: The Frey curve's discriminant \(\Delta = (abc)^{2n}\) and conductor \(N\) relate to the \(j\)-invariant, which encodes lattice symmetries (elliptic curves are complex tori \(\mathbb{C}/\Lambda\), where \(\Lambda\) is a lattice). The modularity theorem connects these lattices to modular forms — a deep symmetry group (SL(2,\(\mathbb{Z}\)) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com