Wiles' Modularity Proof of Fermat's Last Theorem and Its Expository Generalizations — E8 Intelligence Research

FINDING: Fermat's Last Theorem (FLT) — no positive integers a,b,c satisfy aⁿ+bⁿ=cⁿ for n>2 — was proven by Wiles (1995) via the modularity of semistable elliptic curves, not by elementary means; the "generalizations" in these sources are mostly expository, with one arXiv preprint proposing a polynomial-root reformulation. | MATH: FLT: aⁿ + bⁿ = cⁿ has no solutions in ℤ⁺ for n ≥ 3. Wiles' chain: Frey curve E: y² = x(x−aⁿ)(x+bⁿ) → if FLT failed, E would be a non-modular semistable elliptic curve → contradicts Taniyama–Shimura–Weil (modularity theorem): every elliptic curve over ℚ is modular. Key objects: elliptic curves, modular forms, Galois representations ρ: Gal(ℚ̄/ℚ) → GL₂(𝔽ₚ), deformation rings, Hecke algebras. | CONNECTION: The proof's core is a bridge between two symmetry structures — the automorphic (modular forms, SL₂(ℤ) action on the upper half-plane, a discrete subgroup of PSL₂(ℝ)) and the arithmetic (Galois representations). The modular group Γ = SL₂(ℤ) has a fundamental doma Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com