Fermat's Last Theorem

Solved in 1994 by Andrew Wiles (with Richard Taylor).

Around 1637 Fermat wrote in the margin of his copy of Diophantus's Arithmetica that he had a marvelous proof that the margin was too narrow to contain. He did prove the case $n=4$, which reduces the problem to prime exponents, and many further cases followed, but a general proof stayed out of reach for over 350 years. In the 1980s Gerhard Frey, Jean-Pierre Serre and Ken Ribet showed that a counterexample would give an elliptic curve violating the Taniyama–Shimura conjecture, which says that elliptic curves are modular. Andrew Wiles announced a proof of enough of that conjecture in 1993, a gap was found during review, and he repaired it in 1994 with his former student Richard Taylor. The proof was published in 1995.

References

• Fermat's Last Theorem (Wikipedia)