06 July, 2009

Fermat's Last Theorem

I'm no mathematician. However there is something that has always intrigued me about the purity of mathematical proof. That is why I tend to read a lot of popular science books on maths. An excellent example is Simon Singh's book detailing the events behind the proof to Fermat's Last Theorem.

We all know Pythagorus' Theorem, it was drilled into us at school:

a^2 + b^2 = c^2\!\,

This equation is true when c is the hypoteneuse of a triangle and a and b are the other two sides. It's a straightforward enough mathematical concept and has been proved many times in completely different ways. Indeed, as a child, I was taught a number of the more easily understandable methods.

Now...replace the number 2 in the above equation with n>2 (that is any number greater than 2) and replace the equals sign with a 'does not equal' sign. This is known as Fermat's Last Theorem and can be summarised as:

If an integer n is greater than 2, then the equation an + bn = cn has no solutions in non-zero integers a, b, and c.

In principle, this makes the equation no more difficult to understand. Essentially, you can split a square number into a sum of two lower square numbers, but you can't do the same for cube number or higher powers. Easy enough.

Should be simple enough to prove right?


A man called Pierre de Fermat claimed to have a proof in 1637, although bizarrely he didn't record it as he didn't have enough space in the margin of his copy of Arithmetica:

To resolve a cube into the sum of two cubes, a fourth power into two fourth powers, or in general any power higher than the second into two of the same kind, is impossible; of which fact I have found a remarkable proof. The margin is too small to contain it.

There is no doubt that Fermat was a brilliant mathematician in his own right and provided many proofs which were later verified. Thus, the proof he provided that became known as Fermat's Last Theorem was so called as it was the last of Fermat's asserted theorems to remain unproven. This gave it an air of romanticism which attracted many people, all hoping to be the one to finally crack it, but the proof proved elusive.

In the following centuries many people attempted to prove the theorem (technically it was a conjecture, not a theorem) but were unable to do so. Instead they succeeded in proving that there were no solutions for specific integers; Euler provided proof for n=3, Fermat himself for n=4, and so on.

A big step towards a proof came in the 1980s with the realisation that a seemingly unrelated mathematical conjecture, called the Taniyama–Shimura–Weil conjecture, when applied to certain elliptic curves actually implies Fermat's Last Theorem. This is where Andrew Wiles comes in. He had been fascinated by the theorem since childhood and had secretly been attempting his own proof for some time. When he heard about the link with the Taniyama–Shimura–Weil conjecture, he gave up his other commitments and worked on this association for several years in the attic of his house.

I don't want to give away any more of the story as the details can be read in the Simon Singh book Fermat's Last Theorem, also titled as Fermat's Enigma (I recommend it highly), but to summarise, in 1993 Wiles eventually succeeded in proving the theorem, thus fulfilling a life-long dream, although the story didn't end there as there were a few twists and turns after that.

However, the intriguing thing is that in order to prove the theorem, Wiles had to develop several novel mathematical techniques that would be considered as '20th century mathematics' and were thus unavailable to Fermat back in 1637. So the question still remains as to whether Fermat did actually have a proof. It's seems impossible, due to the sophisticated new techniques needed, but it cannot be categorically denied, fueling the intrigue.

Fermat's alleged "marvellous proof"...would have had to be fairly elementary, given the state of the mathematical knowledge at the time, and so could not have been the same as Wiles's. And in fact, most mathematicians and science historians doubt that Fermat had a valid proof of his theorem for all exponents n, as it seems unlikely there is an elementary proof.

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