In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two. This theorem was first conjectured by Pierre de Fermat in 1637, but was not proven until 1995 despite the efforts of many illustrious mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th. It is among the most famous theorems in the history of mathematics.
Fermat left no proof of the conjecture for all n, but he did prove the special case n = 4. This reduced the problem to proving the theorem for exponents n that are odd prime numbers. Over the next two centuries (1637â€“1839), the conjecture was proven for only the primes 3, 5, and 7, although Sophie Germain proved a special case for all primes less than 100. In the mid-19th century, Ernst Kummer proved the theorem for a large (probably infinite) class of primes known as regular primes. Building on Kummer's work and using sophisticated computer studies, other mathematicians were able to prove the conjecture for all odd primes up to four million.
The final proof of the conjecture for all n came in the late 20th century. In 1984 Gerhard Frey suggested the approach of proving the conjecture through the modularity conjecture for elliptic curves. Building on work of Ken Ribet, Andrew Wiles succeeded in proving enough of the modularity conjecture to prove Fermat's Last Theorem, with the assistance of Richard Taylor. Wiles's achievement was reported widely in the popular press, and has been popularized in books and television programs.