Wiles' Modularity Proof of Fermat's Last Theorem — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — no positive integers a,b,c satisfy aⁿ+bⁿ=cⁿ for integer n>2 — was proved by Wiles (1994) via the modularity of semistable elliptic curves, not by elementary number theory. | MATH: FLT: aⁿ + bⁿ = cⁿ has no solutions in ℤ⁺ for n ≥ 3. Wiles–Taylor: every semistable elliptic curve over ℚ is modular (Taniyama–Shimura–Weil conjecture for that class). Frey curve: for a hypothetical solution aᵖ+bᵖ=cᵖ (p odd prime), construct E: y² = x(x−aᵖ)(x+bᵖ), which is semistable with discriminant Δ = (abc)²ᵖ/2⁸ and conductor N = rad(abc). Ribet's level-lowering: such E would be non-modular, contradicting Wiles. | CONNECTION: Modular forms live on the upper half-plane with SL₂(ℤ) symmetry; the modular group Γ = PSL₂(ℤ) ≅ ℤ/2 * ℤ/3, whose generators S (order 2) and T (order 3) generate the (2,3,∞) triangle group — the same (2,3,5) icosahedral / (2,3,4) octahedral / (2,3,6) triangular crystallographic reflection groups appear as finite quotients. The j-invariant j(τ) = Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com