Fermat's Last Theorem (24 page)

Turing's legacy was a machine which could take an unpractically long calculation, if performed by a human, and complete it in a matter of hours. Today's computers perform more calculations in a split second than Fermat performed in his entire career. Mathematicians who were still struggling with Fermat's Last Theorem began to use computers to attack the problem, relying on a computerised version of Kummer's nineteenth-century approach.

Kummer, having discovered a flaw in the work of Cauchy and Lamé, showed that the outstanding problem in proving Fermat's Last Theorem was disposing of the cases when
n
equals an irregular prime – for values of
n
up to 100 the only irregular primes are 37, 59 and 67. At the same time Kummer showed that in theory all irregular primes could be dealt with on an individual basis, the only problem being that each one would require an enormous amount of calculation. To make his point Kummer and his colleague Dimitri Mirimanoff put in the weeks of calculation required to dispel the three irregular primes less than 100. However, they and other mathematicians were not prepared to begin on the next batch of irregular primes between 100 and 1,000.

A few decades later the problems of immense calculation began to vanish. With the arrival of the computer awkward cases of Fermat's Last Theorem could be dispatched with speed, and after the Second World War teams of computer scientists and mathematicians proved Fermat's Last Theorem for all values of
n
up to 500, then 1,000, and then 10,000. In the 1980s Samuel S. Wagstaff of the University of Illinois raised the limit to 25,000 and more recently mathematicians could claim that Fermat's Last Theorem was true for all values of
n
up to 4 million.

Although outsiders felt that modern technology was at last getting the better of the Last Theorem, the mathematical community were aware that their success was purely cosmetic. Even if supercomputers spent decades proving one value of
n
after another they could never prove every value of
n
up to infinity, and therefore they could never claim to prove the entire theorem. Even if the theorem was to be proved true for up to a billion, there is no reason why it should be true for a billion and one. If the theorem was to be proved up to a trillion, there is no reason why it should be true for a trillion and one, and so on
ad infinitum.
Infinity is unobtainable by the mere brute force of computerised number crunching.

David Lodge in his book
The Picturegoers
gives a beautiful description of eternity which is also relevant to the parallel concept of infinity: ‘Think of a ball of steel as large as the world, and a fly alighting on it once every million years. When the ball of steel is rubbed away by the friction, eternity will not even have begun.'

All that computers could offer was evidence in favour of Fermat's Last Theorem. To the casual observer the evidence might seem to be overwhelming, but no amount of evidence is enough to satisfy mathematicians, a community of sceptics who will accept nothing other than absolute proof. Extrapolating a theory to cover an infinity of numbers based on evidence from a few numbers is a risky (and unacceptable) gamble.

One particular sequence of primes shows that extrapolation is a dangerous crutch upon which to rely. In the seventeenth century mathematicians showed by detailed examination that the following numbers are all prime:

31; 331; 3,331; 33,331; 333,331; 3,333,331; 33,333,331.

The next numbers in the sequence become increasingly giant, and checking whether or not they are also prime would have taken
considerable effort. At the time some mathematicians were tempted to extrapolate from the pattern so far, and assume that all numbers of this form are prime. However, the next number in the pattern, 333,333,331, turned out not to be a prime:

Another good example which demonstrates why mathematicians refused to be persuaded by the evidence of computers is the case of Euler's conjecture. Euler claimed that there were no solutions to an equation not dissimilar to Fermat's equation:

For two hundred years nobody could prove Euler's conjecture, but on the other hand nobody could disprove it by finding a counterexample. First manual searches and then years of computer sifting failed to find a solution. Lack of a counter-example was strong evidence in favour of the conjecture. Then in 1988 Naom Elkies of Harvard University discovered the following solution:

Despite all the evidence Euler's conjecture turned out to be false. In fact Elkies proved that there were infinitely many solutions to the equation. The moral is that you cannot use evidence from the first million numbers to prove a conjecture about all numbers.

But the deceptive nature of Euler's conjecture is nothing compared to the
overestimated prime conjecture.
By scouring through larger and larger regimes of numbers, it becomes clear that the prime numbers become harder and harder to find. For instance, between 0 and 100 there are 25 primes but between 10,000,000 and 10,000,100 there are only 2 prime numbers. In 1791, when he was just fourteen years old, Carl Gauss predicted the approximate
manner in which the frequency of prime numbers among all the other numbers would diminish. The formula was reasonably accurate but always seemed slightly to overestimate the true distribution of primes. Testing for primes up to a million, a billion or a trillion would always show that Gauss's formula was marginally too generous and mathematicians were strongly tempted to believe that this would hold true for all numbers up to infinity, and thus was born the overestimated prime conjecture.

Then, in 1914, J.E. Littlewood, G.H. Hardy's collaborator at Cambridge, proved that in a sufficiently large regime Gauss's formula would
underestimate
the number of primes. In 1955 S. Skewes showed that the underestimate would occur sometime before reaching the number

This is a number beyond the imagination, and beyond any practical application. Hardy called Skewes's number ‘the largest number which has ever served any definite purpose in mathematics'. He calculated that if one played chess with all the particles in the universe (10
⁸⁷
), where a move meant simply interchanging any two particles, then the number of possible games was roughly Skewes's number.

There was no reason why Fermat's Last Theorem should not turn out to be as cruel and deceptive as Euler's conjecture or the overestimated prime conjecture.

In 1975 Andrew Wiles began his career as a graduate student at Cambridge University. Over the next three years he was to work
on his Ph.D. thesis and in that way serve his mathematical apprenticeship. Each student was guided and nurtured by a supervisor and in Wiles's case that was the Australian John Coates, a professor at Emmanuel College, originally from Possum Brush, New South Wales.

Coates still recalls how he adopted Wiles: ‘I remember a colleague told me that he had a very good student who was just finishing part III of the mathematical tripos, and he urged me to take him as a student. I was very fortunate to have Andrew as a student. Even as a research student he had very deep ideas and it was always clear that he was a mathematician who would do great things. Of course, at that stage there was no question of any research student starting work directly on Fermat's Last Theorem. It was too difficult even for a thoroughly experienced mathematician.'

For the past decade everything Wiles had done was directed towards preparing himself to meet Fermat's challenge, but now that he had joined the ranks of the professional mathematicians he had to be more pragmatic. He remembers how he had to temporarily surrender his dream: ‘When I went to Cambridge I really put aside Fermat. It's not that I forgot about it – it was always there – but I realised that the only techniques we had to tackle it had been around for 130 years. It didn't seem that these techniques were really getting to the root of the problem. The problem with working on Fermat was that you could spend years getting nowhere. It's fine to work on any problem, so long as it generates interesting mathematics along the way – even if you don't solve it at the end of the day. The definition of a good mathematical problem is the mathematics it generates rather than the problem itself.'

It was John Coates's responsibility to find Andrew a new obsession, something which would occupy his research for at least the next three years. ‘I think all a research supervisor can do for a student is try and push him in a fruitful direction. Of course, it's impossible to be sure what is a fruitful direction in terms of research but perhaps one thing that an older mathematician can do is use his horse sense, his intuition of what is a good area, and then it's really up to the student as to how far he can go in that direction.' In the end Coates decided that Wiles should study an area of mathematics known as
elliptic curves.
This decision would eventually prove to be a turning point in Wiles's career and give him the techniques he would require for a new approach to tackling Fermat's Last Theorem.

The name ‘elliptic curves' is somewhat misleading for they are neither ellipses nor even curved in the normal sense of the word. Rather they are any equations which have the form