Fermat’s Last Theorem: Journey to the Center of Mathematical Genius (and Excess)

1. Fermat’s Last Theorem: Journey to the Center of Mathematical Genius (and Excess) Rod Sanjabi

2. So, who is this Firm-at guy? Pierre de Fermat (Fer’-mah, though some pronounce the ‘t’), a 17th century French mathematician, is thought of today primarily as a number theorist. Ironically, this well-known mathematician was in fact only an amateur in life; he was a lawyer by trade.

3. So, if he’s just an amateur, what’s the big deal? Fermat refused to publish his work, and because of this, 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 and commentaries with the object of publishing a notebook of his ideas. In this way, the famous (or infamous) ’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.

4. OK, so what’s this brilliant theorem? It’s actually pretty simple: xn+ynzn If n>2 Or, in English, the sum of x and y raised to a certain power n (where n is greater than 2) can equal no integer z raised to the same power (as long as x, y, and z are all nonzero integers).

5. Ha! Prove it... Now that, dear heading, is another thing altogether. Fermat’s last theorem (stated before) proved impossible to prove until very recently. Fermat himself said that“I have discovered a truly remarkable proof which this margin is too small to contain”, however, the proof has been long in coming, and after centuries of failed attempts, has only been provided by the use of modern mathematical methods.

6. Does that mean that Fermat was lying? Maybe. Some of the older attempted proofs include an attempt by Euler, who wrote in a letter on 4 August 1753 claiming he had a proof of Fermat's Theorem when n = 3. The proof he provided in Algebra (1770) contains a fallacy and it is far from easy to give an alternative proof of the statement which has the fallacious proof. His mistake, nonetheless, is an interesting one, one which was to have a bearing on later developments. He needed to find cubes of the form p2 + 3q2. He imaginatively attempted to display the proof using numbers of the form a+b(root 3), but he overlooked the fact that such numbers do not behave as integers would.

7. Ok... It doesn’t make all that much sense in plain English, but there were many other notable attempts at proving Fermat’s Last Theorem by celebrated mathematicians including Sophie Germain, Lejeune Dirichlet, Gabriel Lamé (whose proof involved ‘factorizing’ the equation into linear factors over the complex numbers), and others. Most notably, they all failed.

8. And then... With the advent of more and more advanced mathematical principles, proofs for Fermat’s last theorem became more and more complex. Bernoulli’s numbers (defined by x/(ex - 1) = Bn xn /n!) were employed to prove the theorem for all regular prime numbers, but when it was shown that the number of irregular (as relating to Bernoulli’s equation) primes is infinite, the nearly-solved proof became infinitely further from solution (pun intended).

9. So, finally, The final chapter in the story began in 1955, although at the stage the work was not thought of as connected with Fermat's Last Theorem. Yutaka Taniyama asked some questions about elliptic curves, i.e. curves of the form y2=x3+ax+b for constants a and b. Further work by Weil and Shimura produced a conjecture, now known as the Shimura-Taniyama-Weil Conjecture. In 1986 the connection was made between the Shimura-Taniyama-Weil Conjecture and Fermat's Last Theorem by Frey at Saarbrücken showing that Fermat's Last Theorem was far from being some unimportant curiosity in number theory but was in fact related to fundamental properties of space. The proof of Fermat's Last Theorem was completed in 1993 by Andrew Wiles, a British mathematician working at Princeton in the USA. Wiles gave a series of three lectures at the Isaac Newton Institute in Cambridge, England, at the end of which he provided the proof. In view of speculation, Wiles realized that the proof was not completed. Grrr...

10. From the beginning of 1994, Wiles began to collaborate with Richard Taylor in an attempt to fill the holes in the proof. However they decided that one of the key steps in the proof, using methods due to Flach, could not be made to work. They tried a new approach with a similar lack of success. In August 1994 Wiles addressed the International Congress of Mathematicians but was no nearer to solving the difficulties. Taylor suggested a last attempt to extend Flach's method in the way necessary and Wiles, although convinced it would not work, agreed mainly to enable him to convince Taylor that it could never work. Wiles worked on it for about two weeks, then suddenly inspiration struck: “In a flash I saw that the thing that stopped it [the extension of Flach's method] working was something that would make another method I had tried previously work.”

11. In the end... The proof took over 300 years to prove, and elliptic and fractal geometry were used in the final solution...neither of which was present in Fermat’s day. So, did he really have a proof? xn+ynzn... or something.