There are no positive natural numbers a, b, and c such that
in which n is a natural number greater than 2.
The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet's translation of famous Diophantus' Arithmetica, "I have discovered a truly remarkable proof but this margin is too small to contain it". The reason why this statement is so significant is that all the other theorems proposed by Fermat were settled either by proofs he supplied, or by more rigorous proofs supplied afterwards. Mathematicians long were baffled by this statement, for they were unable either to prove or to disprove it. The theorem has the credit of the largest number of wrong proofs.
For various special exponents n, the theorem had been proved over the years, but the general case remained elusive. In 1983 Gerd Faltings proved the Mordell conjecture, which implies that for any n > 2, there are at most finitely many coprime integers a, b and c with an + bn = cn.
Using sophisticated tools from algebraic geometry (in particular elliptic curves and modular forms), Galois theory and Hecke algebras, the English mathematician Andrew Wiles, from Princeton University, with help from his former student Richard Taylor, devised a proof of Fermat's last theorem that was published in 1995 in the journal Annals of Mathematics.
Frey had conjectured ("epsilon conjecture"), and Ribet had proved in 1986, that every counterexample an + bn = cn to Fermat's last theorem would yield an elliptic curve
This latter conjecture proposes a deep connection between elliptic curves and modular forms.
Wiles and Taylor were able to establish a special case of the Taniyama-Shimura conjecture sufficient to exclude such counterexamples arising from Fermat's last theorem.
The story of the proof is almost as remarkable as the mystery of the theorem itself. Wiles spent seven years in isolation working out nearly all the details. When he announced his proof in June 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious problem was discovered: it seemed to lead to the breakdown of this original proof. Wiles and Taylor then spent about a year trying to revive the proof. In September 1994, they were able to resurrect the proof with some different, discarded techniques that Wiles had used in his earlier attempts.
There is considerable doubt over whether Fermat's "truly remarkable proof" was correct. The methods used by Wiles were unknown when Fermat was writing, and it seems inconceivable that Fermat managed to derive all the necessary mathematics to demonstrate the same solution (in the words of Andrew Wiles, "it's impossible; this is a 20th century proof"). The alternatives are that there is a simpler proof that all other mathematicians up until this point have missed, or that Fermat was mistaken. In fact, a plausible faulty proof that might have been accessible to Fermat has been suggested. It is based on the mistaken assumption that unique factorization works in all rings of integral elements of algebraic number fields. The fact that Fermat never published an attempted proof, or even publicly announced that he had one, suggests that he may have found his own error and simply neglected to cross out his marginal note.