What was Fermat's Last Theorem, and how was it proved?

Fermat's Last Theorem states that there are no nonzero whole numbers a, b, c and n, with n greater than 2, such that aⁿ + bⁿ = cⁿ. Andrew Wiles proved it by showing that all semistable elliptic curves over the rational numbers are modular. Earlier work by Frey, Serre and Ribet had already shown that this result implies Fermat's theorem. The corrected proof was published in 1995, 358 years after the problem was conjectured. [1] [2]

The equation xⁿ + yⁿ = zⁿ has plenty of solutions when n is 2. These are the Pythagorean triples, and there are infinitely many of them. The 17th-century mathematician Pierre de Fermat became convinced that no such solutions exist once n exceeds 2. In 1637 he wrote in the margin of his maths textbook that he had a marvellous proof, which the margin was too narrow to contain. That note challenged mathematicians for centuries. [3]

The modularity conjecture grew out of work by Shimura and Taniyama in the 1950s and 1960s. It became widely known through a 1967 paper by Weil, and it says that every elliptic curve over the rationals is modular. In 1985 Frey observed that this conjecture should imply Fermat's Last Theorem. His idea was that a hypothetical solution would produce an elliptic curve that could not be modular. Serre formulated the precise mechanism as the ε-conjecture, and Ribet proved it in the summer of 1986. Ribet's result meant that only the semistable curves needed to be handled. [1]

Almost all mathematicians at the time believed both Fermat's theorem and the modularity theorem to be out of reach of the existing knowledge. [2]

Wiles presented his proof in three lectures at the Isaac Newton Institute in Cambridge, from 21 to 23 June 1993. Few in the audience knew beforehand what he had in store. During refereeing, Nick Katz asked clarifying questions that led Wiles to recognise a gap. The gap was in a critical part of the proof that bounded the order of a particular group, because the Euler system used to extend the Kolyvagin and Flach method was incomplete. Only that part was affected, but without it there was no actual proof. [3] [2]

Wiles spent almost a year trying to fix the gap, first alone and then with his former student Richard Taylor. On 19 September 1994 he had an insight. The reason the Kolyvagin–Flach approach failed directly also meant that his original approach using Iwasawa theory could be made to work if strengthened with ideas from the other method. He submitted two manuscripts in October 1994, one of them written with Taylor. Both were published in the May 1995 issue of the Annals of Mathematics. Together they run to 129 pages, and the mathematical community came to accept the proof as correct in its major components. [2]

Fermat's own claimed proof is not known, and the search results do not say what it might have been. The proof described above relies on modern tools that did not exist in his time. [1]