Fermat's last theorem

What is the real significance of Fermat's Last Theorem?

Mathematics has something of a reputation for being boring. Mathematical theorems scarcely less so. And Mathematicians themselves? Well, let’s just say, it is often a mistake to write about them.

Yet Pythagoras was a mathematician, and led a very interesting life, as was Hypatia, who ended up being hacked to death by Christians in Ancient Alexandria, shortly after discovering some properties of conic sections. So the story of Fermat himself, a French mathematician who left a series of riddlesome problems for others to solve (all on the irritating basis that he had solved them already but was not prepared to explain how) might seem also brisk enough. to almost (but not quite) seem to show that maths can be entertaining!

That's certainly what Simon Singh, a physicist by training, and one of our 'British Science Fascists', thought, when he wrote Fermat’s Last Theorem ¹ intending it seems to take on the task of explaining what everyone agrees is a very, very complicated bit of mathematics - Fermat’s Last Theorem - in very very simple terms to the general public.

The book’s subtitle is ‘The Story of the riddle that confounded the world's greatest minds for 358 years”, which is not quite true but serves as a tempting byline. And Singh wisely starts off with a long digression on Pythagoras and assorted other ancients, and in many ways this is the most interesting part the book.

But then we move on to the ‘solution’ found by contemporary mathematicians in America. And the book enters a long slow trajectory - can we represent that as a polynomial curve? which ends up with pictures of mathematicians at blackboards, or holding glasses of wine at conferences, and ever more obscure ‘semi-explanations’ of things like elliptical curves and modular equations. Do we really need all this? Certainly few if any readers will want it. Singh himself seems to give up somewhere around chapter five after the story has moved on to events in post-war Japan and the Shimura–Taniyama conjecture on ... well the reader can buy the book if they want to know more. After all, we started off being tantalised by the book ‘s subtitle promised, not some obscure conjecture from the 1950s.

So back to the ‘theorem’ itself, as Singh does from time to time throughout the 345 pages including appendices, and it can be explained in a line or two.

Start off with the not too complicated equation the equation is

x² + y² = z²

This, for those of a more scholarly bent, is also Pythagoras’ famous theorem that relates the lengths of the two shorter legs of a right triangle, {x, y}, to the length of the longer leg or hypotenuse, z.

Anyway, by plugging values into x, y and z , we find some tidy looking solutions such as as 3²+4² = 5². (9 + 16 = 25). Another one is x=20, y=21 and z=29. You can try and find the next few, but be warned, the effort will send you mad.

This is fascinating stuff. And so here is the riddle:

Fermat's Last Theorem

xn + yn = zn has no solutions where {x, y, z, n} are positive integers and n > 2.

Fermat wrote this theorem in the margins of his personal copy of Diophantus' Arithmetica. He says he has a tidy proof that shows it too, but as it requires more than the margin of the page he is writing on, he can’t give it. As Singh notes in one of him better asides, some wag recently wrote on the wall in the New York subway : xⁿ + yⁿ = zⁿ - I have found the solution, but as my train is coming I haven’t time to write it here.’

In fact, neither Simon Singh, nor the BBC documentary Horizon that the book is based on, nor the ‘hero’ of the book, mathematician Andrew Wiles (“who had been fascinated by the riddle since childhood”) provide the answer as to ‘what was Fermat’s last theorem’, only a convoluted attempt at proving his assertion is correct. As Singh himself notes, whatever Fermat had in mind, it would not have run to more than one hundred typed pages as Wiles' one did, nor would it have involved the negative ‘imaginary’ parts of complex numbers, the use of elliptical curves and the Kolyvagin–Flach method to mention just a few obscurities from ‘modern’ mathematics. It would not have required these because they all involve certain assumptions, assumptions which Fermat might not have agreed with. Take the use of complex numbers in the theorem, for example. It involves the assumption that there exists a strange number called ‘i’ (for imaginary) which when multiplied by itself remains negative. Nowadays, negative numbers are regularly employed in practical calculations in physics and chemistry, and seem to reflect the ‘real world’ - but if you can start by making funny assumptions like that once - why not another time? Why not make a whole load of assumptions and simplify the mathematical challenge a bit? Wiles' obscurities may have satisfied the committee that awarded the Last Theorem prize, but they surely would not have satisfied Fermat himself. And more to the point, the do not satisfy the reader today.

Here is how Singh, however, finishes his triumphant account under the subheading: 'The Lecture of the Century'.

“’So, by May 1993, I was convinced that I had the whole of Fermat’s Last Theorem in my hands,” recalls Wiles [which obviously, literally he did not, see above]. ‘I still wanted to check the proofs some more but there was a conference which was coming up at the end of June in Cambridge, and I thought that would be a wonderful place to announce the proof - it’s my old home town, and I’d been a graduate student there’...

Immediately after the lecture ended the rumour milll started again with renewed vigour and electronic mail flew around the world. Professor Karl Rubin, a former student of Wiles, reported back to his colleagues in America...


“Hi. Andrew gave his first talk today. He did not announce a proof of Taniyama-Shimura, but he is moving in that direction and he has two more lectures. He is still being very secretive about eh result. My best guess is that he is going to prove that if E is an elliptic curve and Galois representation on the points of order 3 on E satisfied certain hypotheses, then E is modular.”’

If Fermat’s own elegant proof remains a mystery, at least Singh help resolve one question. And the answer is assuredly ‘yes’. Maths can be very dull indeed.
An exuberant Andrew Wiles finally 'proving' Fermat's Last Theorem - for TV anyway. So is that the most significant thing about the proof of Fermat's Last Theorem? Not really. The sinister subtext to Simon Singh's book and the BBC's Horizon program etc. etc. , is that there are these 'final' proofs. It fits in with a general view of mathematics-based science marching forward, tidyng up loose ends and vanquishing uncertainty.

Yet, that mathematical purity is built upon sand. Ever since Kurt Gödel's aptly named 'Incompleteness Theorem' it has been known that the consistency of number theory (or any other deductive system) cannot not be proved using the system itself.

However, it seems that mathematicians today, brought up on the Greek notion of axioms and deductive logic, are still reluctant to accept that even the finest proof is ultimately based on 'unprovable assumptions' rather than anything grander ²