diophantine-equations modular-forms
Theorem: the equation \(x^n\) + \(y^n\) = \(z^n\) has no solutions in positive integers x, y, z when n > 2. Fermat noted in 1637 that he had a marvellous proof; the margin was too narrow to contain it. Proven by Andrew Wiles in 1995 via the modularity of semistable elliptic curves.
Written by Fermat in his copy of Diophantus around 1637 with the fatal words "I have discovered a truly marvellous proof of this, which this margin is too narrow to contain." Euler settled n = 3; Dirichlet and Legendre n = 5; Kummer built ideal theory reaching all regular exponents. Sophie Germains partial programme anticipated modern approaches. In September 2026 the proof gained a second kind of certainty: Claude, working largely autonomously over 11 days on the Prove2Me platform, produced the first complete computer-checked formalization — about 13 million lines of Lean, 29,500 intermediate theorems, following the Darmon–Diamond–Taylor exposition and checked by Lean's kernel plus an independent second kernel. No new mathematics; the first end-to-end machine verification of a proof at this scale, building on Mathlib and Buzzard's community FLT project.
Proved by Andrew Wiles — announced June 1993, repaired with Richard Taylor, published 1995. The proof proves the Taniyama–Shimura modularity conjecture for semistable elliptic curves: FLT follows because a counterexample would yield a non-modular semistable curve (Frey's 1985 observation, Ribet's 1986 theorem). Seven years of secret work; the 1993 gap took another eighteen months. The archetypal century-scale human conquest — and the template for how a single mind with the right structures can end a 358-year story.
How to check it: two Annals papers — Wiles (1995) plus Taylor–Wiles on the ring-theoretic lifting step. The Taylor–Wiles method outgrew FLT itself: pushed to full modularity (Breuil–Conrad–Diamond–Taylor 2001), it became the engine room of modern number theory. See the modularity theorem entry here.
Verified by curator — see the API for full claim provenance.
