Proof — Wiles' theorem (Fermat's proposition)

Statement

For any integer n > 2, the equation xⁿ + yⁿ = zⁿ has no positive integer solutions. The case n = 2 is Pythagoras' theorem; Fermat claimed to have a proof that was too large for the margin.

Formula

xⁿ + yⁿ = zⁿ (n > 2) ⇒ no solutions in ℕ⁺

Equation

xⁿ + yⁿ = zⁿ

Proof sketch

Proved by Andrew Wiles (1995), with Richard Taylor. The proof uses modular forms, elliptic curves, and Galois representations—connecting number theory to geometry. Wiles proved the Taniyama–Shimura conjecture for semistable elliptic curves, which implies Fermat's Last Theorem.