Erica Klarreich in Quanta:
When Andrew Wiles proved Fermat’s Last Theorem in the early 1990s, his proof was hailed as a monumental step forward not just for mathematicians but for all of humanity. The theorem is simplicity itself — it posits that xn + yn = zn has no positive whole-number solutions when n is greater than 2. Yet this simple claim tantalized legions of would-be provers for more than 350 years, ever since the French mathematician Pierre de Fermat jotted it down in 1637 in the margin of a copy of Diophantus’ Arithmetica. Fermat, notoriously, wrote that he had discovered “a truly marvelous proof, which this margin is too narrow to contain.” For centuries, professional mathematicians and amateur enthusiasts sought Fermat’s proof — or any proof at all.
The proof Wiles finally came up with (helped by Richard Taylor) was something Fermat would never have dreamed up. It tackled the theorem indirectly, by means of an enormous bridge that mathematicians had conjectured should exist between two distant continents, so to speak, in the mathematical world. Wiles’ proof of Fermat’s Last Theorem boiled down to establishing this bridge between just two little plots of land on the two continents. The proof, which was full of deep new ideas, set off a cascade of further results about the two sides of this bridge.
More here.