Fermat's Last Theorem (18 page)

After the breakthrough of Sophie Germain the French Academy of Sciences offered a series of prizes, including a gold medal and 3,000 Francs to the mathematician who could finally put to rest the mystery of Fermat's Last Theorem. As well as the prestige of proving Fermat's Last Theorem there was now an immensely valuable reward attached to the challenge. The salons of Paris were full of rumours as to who was adopting which strategy and how close they were to announcing a result. Then, on 1 March 1847, the Academy held its most dramatic meeting ever.

The proceedings describe how Gabriel Lamé, who had proved the case
n
= 7 some years earlier, took the podium in front of the most eminent mathematicians of the age and proclaimed that he was on the verge of proving Fermat's Last Theorem. He admitted that his proof was still incomplete, but he outlined his method and predicted with relish that he would in the coming weeks publish a complete proof in the Academy's journal.

The entire audience was stunned, but as soon as Lamé left the floor Augustin Louis Cauchy, another of Paris's finest mathematicians, asked for permission to speak. Cauchy announced to the Academy that he had been working along similar lines to Lamé, and that he too was about to publish a complete proof.

Both Cauchy and Lamé realised that time was of the essence. Whoever would be first to submit a complete proof would receive the most prestigious and valuable prize in mathematics. Although neither of them had a complete proof, the two rivals were keen to somehow stake a claim and so just three weeks after they had made their announcements they deposited sealed envelopes at the Academy. This was a common practice at the time which enabled mathematicians to go on record without revealing the exact details of their work. If a dispute should later arise regarding the originality of ideas, then a sealed envelope would provide the evidence needed to establish priority.

The anticipation built up throughout April as Cauchy and Lamé published tantalising but vague details of their proof in the proceedings of the Academy. Although the entire mathematical community was desperate to see the proof completed, many of them secretly hoped that it would be Lamé and not Cauchy who would win the race. By all accounts Cauchy was a self-righteous creature, a religious bigot and extremely unpopular with his colleagues. He was only tolerated at the Academy because of his brilliance.

Then, on 24 May, an announcement was made which put an end to the speculation. It was neither Cauchy nor Lamé who addressed the Academy but rather Joseph Liouville. Liouville shocked the entire audience by reading out the contents of a letter from the German mathematician Ernst Kummer.

Kummer was a number theorist of the highest order, but for much of his career a fierce patriotism fired by a hatred of Napoleon deflected him from his true calling. When Kummer was an infant the French army invaded his home town of Sorau, bringing with them an epidemic of typhus. Kummer's father was the town physician and within weeks he was taken by the disease. Traumatised by the experience Kummer swore to do his utmost to defend his country from further attack, and as soon as he left university he applied his intellect to the problem of plotting the trajectories of cannon-balls. Ultimately he taught the laws of ballistics at Berlin's war college.

In parallel with his military career Kummer actively pursued pure mathematical research and had been fully aware of the ongoing saga at the French Academy. He had read through the proceedings and analysed the few details that Cauchy and Lamé had dared to reveal. To Kummer it was obvious that the two Frenchmen were heading towards the same logical dead end, and he outlined his reasons in the letter which he sent to Liouville.

According to Kummer the fundamental problem was that the proofs of both Cauchy and Lamé relied on using a property of numbers known as unique factorisation. Unique factorisation states that there is only one possible combination of primes which will multiply together to give any particular number. For instance, the only combination of primes which will build the number 18 is as follows:

Similarly, the following numbers are uniquely factorised in the following ways:

Unique factorisation was discovered back in the fourth century
BC
by Euclid, who proved that it is true for all counting numbers and described the proof in Book IX of his
Elements.
The fact that unique factorisation is true for all counting numbers is a vital element in many other proofs and is nowadays called the
fundamental theorem of arithmetic.

At first sight there should have been no reason why Cauchy and Lamé should not rely on unique factorisation, as had hundreds of mathematicians before them. Unfortunately both of their proofs involved imaginary numbers. Although unique factorisation is true for real numbers, Kummer pointed out that it might not necessarily hold true when imaginary numbers are introduced. According to him this was a fatal flaw.

For example, if we restrict ourselves to real numbers then the number 12 can only be factorised into 2 × 2 × 3. However, if we allow imaginary numbers into our proof then 12 can also be factorised in the following way:

Here (1 + √–11) is a complex number, a combination of a real and an imaginary number. Although the process of multiplication is more convoluted than for ordinary numbers, the existence of complex numbers does lead to additional ways to factorise 12. Another way to factorise 12 is (2 + √–8) × (2 – √–8). There is no longer a unique factorisation but rather a choice of factorisations.

This loss of unique factorisation severely damaged the proofs of Cauchy and Lamé, but it did not necessarily destroy them completely. The proofs were supposed to show that there were no solutions to the equation
x
ⁿ
+
y
ⁿ
=
z
ⁿ
, where
n
represents any number greater than 2. As discussed earlier in this chapter, the proof only had to work for the prime values of
n.
Kummer showed that by employing extra techniques it was possible to restore unique factorisation for various values of
n.
For example, the problem of unique factorisation could be circumvented for all prime numbers up to and including
n
= 31. However, the prime number
n
= 37 could not be dealt with so easily. Among the other primes less than 100, two others,
n
= 59 and 67, were also awkward cases. These so-called
irregular primes, which are sprinkled throughout the remaining prime numbers, were now the stumbling block to a complete proof.

Kummer pointed out that there was no known mathematics which could tackle all these irregular primes in one fell swoop. However, he did believe that, by carefully tailoring techniques to each individual irregular prime, they could be dealt with one by one. Developing these customised techniques would be a slow and painful exercise, and worse still the number of irregular primes is still infinite. Disposing of them individually would occupy the world's community of mathematicians until the end of time.

Kummer's letter had a devastating effect on Lamé. With hindsight the assumption of unique factorisation was at best over-optimistic and at worst foolhardy. Lamé realised that had he been more open about his work he might have spotted the error sooner, and he wrote to his colleague Dirichlet in Berlin: ‘If only you had been in Paris, or I had been in Berlin, all of this would not have happened.'

While Lamé felt humiliated, Cauchy refused to accept defeat. He felt that compared to Lamé's proof his own approach was less reliant on unique factorisation, and until Kummer's analysis had been fully checked there was the possibility that it was flawed. For several weeks he continued to publish articles on the subject, but by the end of the summer he too fell silent.

Kummer had demonstrated that a complete proof of Fermat's Last Theorem was beyond the current mathematical approaches. It was a brilliant piece of mathematical logic, but a massive blow to an entire generation of mathematicians who had hoped that they might solve the world's hardest mathematical problem.

The situation was summarised by Cauchy, who in 1857 wrote the Academy's closing report on their prize for Fermat's Last Theorem:

Report on the competition for the Grand Prize in mathematical sciences. Already set in the competition for 1853 and prorogued to 1856.

Eleven memoirs have been presented to the secretary. But none has solved the proposed question. Thus, after many times being put forward for a prize, the question remains at the point where Monsieur Kummer left it. However, the mathematical sciences should congratulate themselves for the works which were undertaken by the geometers, with their desire to solve the question, specially by Monsieur Kummer; and the Commissaries think that the Academy would make an honorable and useful decision if, by withdrawing the question from the competition, it would adjugate the medal to Monsieur Kummer, for his beautiful researches on the complex numbers composed of roots of unity and integers.

For over two centuries every attempt to rediscover the proof of Fermat's Last Theorem had ended in failure. Throughout his teenage years Andrew Wiles had studied the work of Euler, Germain, Cauchy, Lamé and finally Kummer. He hoped he could learn by their mistakes, but by the time he was an undergraduate at the University of Oxford he confronted the same brick wall that faced Kummer.

Some of Wiles's contemporaries were beginning to suspect that the problem might be impossible. Perhaps Fermat had deceived himself and therefore the reason why nobody had rediscovered Fermat's proof was that no such proof existed. Despite this scepticism Wiles continued to search for a proof. He was inspired by the knowledge that there had been several cases in the past of proofs which had eventually been discovered only after centuries of effort. And in some of those cases the flash of insight which solved the problem did not rely on new mathematics; rather it was a proof which could have been done long ago.

Figure 11. In these diagrams every dot is connected to every other dot by straight lines. Is it possible to construct a diagram such that every line has at least three dots on it?