 | Fermat's last theorem: Encyclopedia II - Fermat's last theorem - History
Fermat's last theorem - History
Fermat's last theorem - Fermat's comment in the Arithmetica
In problem II.8 of his Arithmetica, Diophantus asks how to split a given square number into two other squares (in modern notation, given a rational number k, find u and v, both rational, such that k2 = u2 + v2), and shows how to solve the problem for k = 4. Around 1640, Fermat wrote the following comment (in Latin) in the margin of this problem in his copy of the Arithmetica:
Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos,
et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem
nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi.
Hanc marginis exiguitas non caperet.
(It is impossible to separate a cube into two cubes, or a fourth power into two
fourth powers, or in general, any power higher than the second into two like
powers. I have discovered a truly marvelous proof of this, which this margin
is too narrow to contain.)
In modern notation, this comment corresponds to the theorem mentioned above. Fermat's copy of the Arithmetica has not been found so far; however, around 1670, his son produced a new edition of the book augmented with comments made by his father, including the comment above which will be known later as Fermat's last theorem.
In the case n = 2, it was already known by Ancient Chinese, Greeks and Babylonians that the Diophantine equation a2 + b2 = c2 (linked with the Pythagorean theorem) has integer solutions, such as (3,4,5) (32 + 42 = 52) or (5,12,13). These solutions are known as Pythagorean triples, and there exists infinitely many of them, even excluding trivial solutions for which a, b and c have a common divisor. Fermat's last theorem is a generalisation of this result to higher powers n, and states that no such solution exists when the exponent 2 is replaced by a larger integer.
Fermat's last theorem - Early history
The theorem needs only to be proven for n=4 and in the cases where n is an odd prime number.[2] For various special exponents n, the theorem had been proven over the years, but the general case remained elusive.
Fermat himself proved the case n=4, while Euler proved the theorem for n=3. The case n=5 was proved by Dirichlet and Legendre in 1825, and the case n=7 by Gabriel Lamé in 1839.
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.
Fermat's last theorem - The proof
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.
In 1986, Ken Ribet had proved Gerhard Frey's epsilon conjecture that every counterexample an + bn = cn to Fermat's last theorem would yield an elliptic curve defined as:
which would provide a counterexample to the Taniyama-Shimura conjecture.
This latter conjecture proposes a deep connection between elliptic curves and modular forms.
Andrew Wiles and Richard 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 working out nearly all the details by himself and with utter secrecy (except for a final review stage for which he enlisted the help of his Princeton colleague, Nick Katz). When he announced his proof over the course of three lectures delivered at Cambridge University on June 21-23 1993, he amazed his audience with the number of ideas and constructions used in his proof. Unfortunately, upon closer inspection a serious error 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.
Other related archives1637, 17th-century, 1825, 1839, 1983, 1993, 1994, 20th century, Academic publishing, Andrew Wiles, Annals of Mathematics, Arcadia, Arthur Porges, Bedazzled, Bell, Eric T., Cambridge University, Captain Picard, Chris Thompson, Claude-Gaspar Bachet, Devil, Diophantine equation, Diophantus, Dirichlet, Euler, Euler's conjecture, Facets, Fermat's little theorem, Gabriel Lamé, Galois theory, Gerd Faltings, Gerhard Frey, Hecke algebras, Jadzia Dax, Ken Ribet, Latin, Legendre, Mathematical theorems, Mordell conjecture, Nick Katz, Number theory, Pierre de Fermat, Princeton University, Pythagorean theorem, Pythagorean triples, Richard Taylor, Simpsons, Singh, Simon, Sophie Germain prime, Star Trek: Deep Space Nine, Star Trek: The Next Generation, Taniyama-Shimura conjecture, The Royale, Tobin Dax, Tom Stoppard, Treehouse of Horror VI, Wall-Sun-Sun prime, algebraic geometry, algebraic number fields, coprime, divisor, eighteenth century, elliptic curves, history of mathematics, hours, integer, integers, integral, lectures, mathematician, modular forms, natural numbers, number theory, prime number, proof, rings, theorems, unique factorization
 Adapted from the Wikipedia article "History", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
