Fermat's Last Theorem

Fermat's last theorem is one of the most famous theorems of mathematics. It was first stated by the brilliant amateur mathematician Pierre de Fermat, but it wasn't proved until more than 350 years later. Over the centuries it provided a tantalizing challenge to great mathematicians and amateurs alike, but it did not yield to proof until 1994, when a proof by Princeton mathematician Andrew Wiles was accepted by the mathematical community as correct.

Although it turned out to be one of the most challenging problems in mathematics, Fermat's last theorem is exceptionally easy to state. It says that whenever n is a whole number bigger than two, the equation x^n + y^n = z^n has no non-zero solutions x, y, and z among the whole numbers.

Fermat's statement was found by his son as a marginal note that the great mathematician had made in his volume of Diophantus's Arithmetica. In the margin Fermat wrote the above statement, then words that have become notorious among mathematicians: "I have discovered a truly remarkable proof which this margin is too small to contain." Since that time, mathematicians have speculated whether Fermat was really able to prove the theorem, or whether he was mistaken or bluffing. Mathematicians agree that the techniques employed by Wiles in his proof are so modern that they could not possibly have been conceived of by Fermat. To this day there are mathematicians who try to find a proof that uses only techniques that could have been known to Fermat.

In the first two centuries that followed Fermat's statement, many eminent mathematicians made progress toward proving Fermat's theorem, but in agonizingly small steps. Fermat's theorem is a statement about all possible exponents n, and all possible solutions x, y, and z. Therefore, it is not possible to check the statement by brute force: even if mathematicians were to test all the numbers x, y, and z less than a billion, say, with all the exponents less than a trillion, and none of them satisfied the equation, there could still be a solution involving larger numbers. Over the years, computers were used to test larger and larger numbers, and their evidence greatly strengthened mathematicians' belief in the truth of the theorem, but it did not blind them to the fact that no amount of checking of specific numbers can constitute a proof.

The first progress towards a proof was made by Leonhard Euler, who claimed in a letter to Christian Goldbach in 1753 that he had proved the theorem for the exponent n=3. The proof that he gave in his book Algebra in 1770 has a fallacy that is not easy to correct, but that is possible to correct using other methods from his book.

The next step towards a proof was made by Sophie Germain, who proved that when n and 2n+1 are both prime numbers, potential solutions x, y, and z could only be of a few narrowly restricted types. Using Germain's idea, mathematicians were able to prove that there are no solutions for several more exponents n.

Germain's discovery led to a flurry of new interest in Fermat's theorem, and the French Academy of Sciences offered a series of prizes to anyone who could come up with a correct proof of the theorem. On March 1, 1847 the academy had one of its most exciting meetings ever, when Gabriel Lame, a highly respected member, informed the academy that he was on the verge of proving Fermat's theorem, and gave a rough outline of his plan of attack. No sooner had he finished than the great mathematician Augustin Louis Cauchy rose to announce that he himself was also near to a proof of the theorem, following somewhat similar lines to those of Lame. A race was on between the two mathematicians, and three weeks later each of them delivered a sealed envelope to the academy detailing their techniques; this was a common proceeding at the time, by which mathematicians could establish priority for a result without having to make their techniques public before they were ready. But on May 24, the hopes of the two mathematicians and the entire academy were dashed when they received a letter from the German mathematician Ernst Kummer. He wrote to point out that both Lame and Cauchy were in error, because they were relying on unique factorization, the fact that any number can be written as a product of primes in a unique way (for example, 18=2x3x3). While this is true of whole numbers, Kummer realized that it was not true of complex numbers, numbers that involve the square root of -1. Lame and Cauchy had been assuming unique factorization of certain kinds of complex numbers, so neither of their proofs was correct.

In 1908, the amateur mathematician Paul Wolfskehl, who had always been fascinated by Fermat's last theorem, bequeathed a prize of one hundred thousand marks to whoever could prove the theorem. This generous prize greatly increased public awareness of the problem, with the result that the University of Göttingen, which was to administer the prize, was deluged with attempts at proofs. Eventually, Edmund Landau, the head of the mathematics department, resorted to sending printed cards acknowledging submissions and stating on which page and line the first error occurred, as it unfailingly did. Even after World War I, when inflation severely damaged the value of the prize, countless seekers of glory tried to prove the theorem.

Substantial progress occurred in 1955 when the Japanese mathematicians Yutaka Taniyama and Goro Shimura did seminal work on elliptic curves, solutions to equations of the form y^2 = x^3 + ax + b, where a and b are constants. Although their work was halted tragically by Taniyama's suicide in 1958, they produced a conjecture that was later shown by Gerhard Frey to imply Fermat's theorem. In other words, if someone could prove the extremely difficult Taniyama-Shimura conjecture, Fermat's theorem would be proved as well.

The person who succeeded in proving the Taniyama-Shimura conjecture was Andrew Wiles, who had been fascinated by Fermat's last theorem since childhood. In order to have complete concentration and not to be scooped by competitors, Wiles shut himself up in an office inside his home and worked steadily on the theorem for seven years, without telling his colleagues what he was doing. Finally in 1993, amid rumors that something dramatic was to occur, Wiles gave a series of lectures on his work at the Isaac Newton Institute. At the end of the last lecture Wiles wrote Fermat's Last Theorem on the blackboard as a consequence of his work, saying "I think I'll stop here," to tumultuous applause.

The voluminous proof still had to be subjected to peer review, and during that process an error was discovered in the proof, which Wiles was at first unable to correct. He began to collaborate with Richard Taylor to try to fill the gap, and after an anxious year, Wiles realized that he could correct the error using an extension of a technique of Matheus Flach that Taylor had brought to his attention. With that insight he was in fact able to simplify the proof, and the current proof as it stands is generally accepted as correct, bringing to a close one of the most dramatic stories of mathematics.