Fermat's Last Theorem: Elliptic Proofs and a New Elementary Polynomial Path — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — xⁿ + yⁿ = zⁿ has no integer solutions for n > 2 — is proven via elliptic curves and modular forms, with a novel polynomial-root approach proposed for elementary access. | MATH: Core equation: aⁿ + bⁿ = cⁿ, n ∈ ℤ, n > 2 → no positive integer solutions. Wiles' proof: every semistable elliptic curve over ℚ is modular (Taniyama–Shimura–Weil). Key objects: Frey curve E: y² = x(x − aⁿ)(x + bⁿ), discriminant Δ = (abc)²ⁿ/16, conductor N. Modularity: ρ_E,p ≅ ρ_f,p for a weight-2 newform f. Polynomial approach (arXiv:1105.0669): associate P(x) = (x + a)ⁿ + (x + b)ⁿ − (x + c)ⁿ; roots studied to infer FLT validity. | CONNECTION: The Frey curve's discriminant Δ = (abc)²ⁿ/16 contains the factor 16 = 2⁴, linking to 2-adic structure and base-2, not directly base-60. However, the modular form's weight 2 and level N relate to the Eisenstein series E₂ and the lattice Λ = ℤ + τℤ — a 2D lattice with fundamental parallelogram area Im(τ). The ratio of lattice periods (τ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com