The story behind Fermat's Last Theorem has exciting elements not usually found in mathematical quests. There is genius, passion, jealousy, money (the Wolfskehl 100,000 DM Prize) and even sex discrimination (in the 19th century, Sophie Germain had to assume a male identity to work on the problem and correspond with other mathematicians).
The formulation is deceptively simple: "prove that there are no whole numbers X, Y and Z such that Xn+Yn=Zn for n > 2" (contrary to n=2 which is the familiar Pythagorean case). It first appeared in Pierre Fermat's notes published in 1670 and was finally solved by Andrew Wiles in 1995. Fermat claimed to "have a truly marvelous demonstration of this proposition which this margin is too narrow to contain". If it were so, certainly it was not the kind of approach which was finally used by Wiles who had to unify all sorts of modern mathematics - click here to see a diagrammatic representation of the final steps to the proof. This is the reason many people (mostly amateurs) are still trying to find a simpler solution with mathematics of Fermat's times. One argument in favor of the existence of such a solution is that all other Fermat's theorems were shown to be true soon after their publication (hence the name Fermat's "Last" Theorem).
The staff in the mathematics department at Göttingen were certainly relieved to award the Wolfskehl prize to Wiles and finish with this story. To get an idea of what was happening, Prof. Edmund Landau who was responsible for the contest entries at the start of the century used to print cards that read:
Dear ... ,
Thank you for your manuscript on the proof
of Fermat's Last Theorem.
The first mistake is on: Page ... Line ...
This invalidates the proof.
Professor E. M. Landau
and passed them to his students as exercises.
The value of this theorem lies more in the discoveries made and the new mathematics created as a by-product of efforts to prove it than in the problem itself. This had been known long ago. In his lecture to the International Congress of Mathematics at Paris in 1900, David Hilbert said:
"The attempt to prove ... offers a striking example of the inspiring effect which such a very special and apparently unimportant problem may have upon science. For Kummer, incited by Fermat's problem, was led to the introduction of ideal numbers and to the discovery of the law of the unique decomposition of the numbers of a circular field into ideal prime factors - a law which to-day, in its generalization to any algebraic field by Dedekind and Kronecker, stands at the center of the modern theory of numbers and whose significance extends far beyond the boundaries of number theory into the realm of algebra and the theory of functions".
Why did it take so long to be solved, in
spite of so many brilliant minds involved (Euler, Germain, Cauchy, Lamé,
Kummer, to name a few)? Answers from experts through Scientific American:
"I would ask this question the other way around. How did we get so lucky as to find a proof at all? The German polymath Karl Gauss summed up the attitudes of many pre-1985 professional mathematicians when in 1816 he wrote: 'I confess that Fermat's Last Theorem, as an isolated proposition, has very little interest for me, because I could easily lay down a multitude of such propositions, which one could neither prove nor dispose of.' Somehow we got lucky and managed to save Fermat's Last Theorem from its isolation by linking it to some important branches of modern mathematics, especially the theory of modular forms. Was this really just luck? How many others of Gauss's 'multitude of propositions' can also be magically transformed and made accessible to the powerful tools of modern mathematics? Fermat's Last Theorem is just the beginning. There are many fascinating explorations still ahead of us!" (Glenn H. Stevens in the mathematics department at Boston University).
"The other problem is that Fermat's
claim has always felt, well, marginal. It is hard to connect the Last Theorem
to other parts of mathematics, which means that powerful mathematical ideas
can't necessarily be applied to it. In fact, if one looks at the history
of the theorem, one sees that the biggest advances in working toward a
proof have arisen when some connection to other mathematics was found.
For example, Polish mathematician Ernst Eduard Kummer's work in the mid-19th
century arises from connecting the Last Theorem to the theory of cyclotomic
fields. And Wiles is no exception: his proof grows out of work by Frey,
Serre and Ribet that connects Fermat's statement with the theory of elliptic
curves. Once that connection was established, and one knew that proving
the Modularity Conjecture for elliptic curves would yield a proof of Fermat's
Last Theorem, there was reason to be hopeful. Wiles's work shows that such
hope was justified." (Fernando Q. Gouve, chair of the department of
mathematics and computer science at Colby College).
The MacTutor History of Mathematics archive
has an excellent page on the history of this puzzle with links and bibliography
- including links to the mathematics of Wiles' solution, for the brave
:) Click here
See also:
A simpler solution here
Famous Problems
and Proofs Main Page