One of the most famous theorems in the history of mathematics with, in order of appearance : Pythagoras, Pierre de Fermat, Leonard Euler, Sophie Germain, Evariste Galois, Paul Wolfskhel, Yutaka Taniyama and Andrew Wiles.
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.
Pierre de Fermat died in 1665. Today we think of Fermat as a number theorist, in fact as perhaps the most famous number theorist who ever lived. It is therefore surprising to find that Fermat was in fact a lawyer and only an amateur mathematician. Also surprising is the fact that he published only one mathematical paper in his life, and that was an anonymous article written as an appendix to a colleague's book.
Because Fermat refused to publish his work, his friends feared that it would soon be forgotten unless something was done about it. His son, Samuel undertook the task of collecting Fermat's letters and other mathematical papers, comments written in books, etc. with the object of publishing his father's mathematical ideas. In this way the famous 'Last theorem' came to be published. It was found by Samuel written as a marginal note in his father's copy of Diophantus's Arithmetica.
Fermat's Last Theorem states that xn + yn = zn has no non-zero integer solutions for x, y and z when n > 2. Fermat wrote "I have discovered a truly remarkable proof which this margin is too small to contain".
Fermat almost certainly wrote the marginal note around 1630, when he first studied Diophantus's Arithmetica. It may well be that Fermat realised that his remarkable proof was wrong, however, since all his other theorems were stated and restated in challenge problems that Fermat sent to other mathematicians. Although the special cases of n = 3 and n = 4 were issued as challenges (and Fermat did know how to prove these) the general theorem was never mentioned again by Fermat.
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.
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