Wiles proves Fermat's Last Theorem

In September 1994 the British mathematician Andrew Wiles, at Princeton University, and his former student Richard Taylor completed the proof of a statement that Pierre de Fermat had scribbled around 1637: for any whole number n above 2, no positive whole numbers a, b and c satisfy a^n + b^n = c^n. Fermat wrote in the margin of his copy of Diophantus's Arithmetica that he had a marvelous proof the margin was too small to hold, and the claim was published after his death. Wiles worked mostly in secret for about seven years, announced a proof in Cambridge in June 1993, and fixed a gap found in review with Taylor's help. The result appeared in the Annals of Mathematics in 1995.¹²⁴

The route was indirect. In 1984 Gerhard Frey noticed that a hypothetical solution to Fermat's equation would yield an elliptic curve with strange properties, and in 1986 Kenneth Ribet proved that such a curve could not be modular. So proving the Taniyama-Shimura modularity conjecture, which says every elliptic curve over the rational numbers is tied to a modular form, would settle Fermat. Wiles proved enough of it, the semistable case, using Galois representations and tools that grew out of the number theory begun by Diophantus, Gauss and Galois. The full modularity theorem followed in 1999.¹³⁴