Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers
can satisfy the equation
for
(n=2 being Pythagoras Theorem for right angled triangles). This theorem was first conjectured by Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin.
Fermat left no proof of the conjecture for all n, but he did prove the special cas
(This case had already been proved by Leonardo Fibonacci in 1225 in his Liber quadratorum although this fact is often overlooked in discussions of Fermat's Last Theorem.) This reduced the problem to proving the theorem for exponents n that are prime numbers. Over the next two centuries (1637–1839), 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.
