by Simon Singh
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 xn + yn = zn, 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?
One example of a problem which evaded solution for decades is the dot conjecture. The challenge involves a series of dots which are all connected to each other by straight lines, such as the dot diagrams shown in Figure 11. The conjecture claims that it is impossible to draw a dot diagram such that every line has at least three dots on it (excluding the diagram where all the dots are on the same line). Certainly by experimenting with a few diagrams this appears to be true. For example, Figure 11(a) has five dots connected by six lines. Four of the lines do not have three dots on them and so clearly this arrangement does not satisfy the requirement that all lines have three dots. By adding an ex
tra dot and the associated line, as in Figure 11(b), the number of lines which do not have three dots is reduced to just three. However, trying to adapt the diagram further so that all the lines have three dots appears to be impossible. Of course, this does not prove that no such diagram exists.
Generations of mathematicians tried and failed to find a proof of the apparently straightforward dot conjecture. What made the conjecture even more infuriating was that when a proof was eventually discovered, it involved only a minimal amount of mathematical knowledge mixed with a little extra cunning. The proof is outlined in Appendix 6.
There was a possibility that all the techniques required to prove Fermat’s Last Theorem were available, and that the only missing ingredient was ingenuity. Wiles was not prepared to give up: finding a proof of the Last Theorem had turned from being a childhood fascination in to a fully fledged obsession. Having learnt all there was to learn about the mathematics of the nineteenth century, Wiles decided to arm himself with techniques of the twentieth century.
4
Into Abstraction
Proof is an idol before which the mathematician tortures himself.
Sir Arthur Eddington
Following the work of Ernst Kummer, hopes of finding a proof for the Last Theorem seemed fainter than ever. Furthermore mathematics was beginning to move into different areas of study and there was a risk that the new generation of mathematicians would ignore what seemed an impossible dead-end problem. By the beginning of the twentieth century the problem still held a special place in the hearts of number theorists, but they treated Fermat’s Last Theorem in the same way that chemists treated alchemy. Both were foolish romantic dreams from a past age.
Then in 1908 Paul Wolfskehl, a German industrialist from Darmstadt, gave the problem a new lease of life. The Wolfskehl family were famous for their wealth and their patronage of the arts and sciences, and Paul was no exception. He had studied mathematics at university and, although he devoted most of his life to building the family’s business empire, he maintained contact with professional mathematicians and continued to dabble in number theory. In particular Wolfskehl refused to give up on Fermat’s Last Theorem.
Wolfskehl was by no means a gifted mathematician and he was not destined to make a major contribution to finding a proof of the Last Theorem. Nonetheless, thanks to a curious chain of events, he was to become forever associated with Fermat’s problem, and would inspire thousands of others to take up the challenge.
The story begins with Wolfskehl’s obsession with a beautiful woman, whose identity has never been established. Depressingly for Wolfskehl the mysterious woman rejected him and he was left in such a state of utter despair that he decided to commit suicide. He was a passionate man, but not impetuous, and he planned his death with meticulous detail. He set a date for his suicide and would shoot himself through the head at the stroke of midnight. In the days that remained he settled all his outstanding business affairs, and on the final day he wrote his will and composed letters to all his close friends and family.
Wolfskehl had been so efficient that everything was completed slightly ahead of his midnight deadline, so to while away the hours he went to the library and began browsing through the mathematical publications. It was not long before he found himself staring at Kummer’s classic paper explaining the failure of Cauchy and Lamé. It was one of the great calculations of the age and suitable reading for the final moments of a suicidal mathematician. Wolfskehl worked through the calculation line by line. Suddenly he was startled at what appeared to be a gap in the logic – Kummer had made an assumption and failed to justify a step in his argument. Wolfskehl wondered whether he had uncovered a serious flaw or whether Kummer’s assumption was justified. If the former were true, then there was a chance that proving Fermat’s Last Theorem might be a good deal easier than many had presumed.
He sat down, explored the inadequate segment of the proof, and became engrossed in developing a mini-proof which would either consolidate Kummer’s work or prove that his assumption was wrong, in which case all Kummer’s work would be invalidated. By dawn his work was complete. The bad news, as far as mathematics was concerned, was that Kummer’s proof had been remedied and the Last Theorem remained in the realm of the unattainable. The good news was that the appointed time of the suicide had passed, and Wolfskehl was so proud that he had discovered and corrected a gap in the work of the great Ernst Kummer that his despair and sorrow evaporated. Mathematics had renewed his desire for life.
Wolfskehl tore up his farewell letters and rewrote his will in the light of what had happened that night. Upon his death in 1908 the new will was read out, and the Wolfskehl family were shocked to discover that Paul had bequeathed a large proportion of his fortune as a prize to be awarded to whomsoever could prove Fermat’s Last Theorem. The reward of 100,000 Marks, worth over £1,000,000 in today’s money, was his way of repaying a debt to the conundrum that had saved his life.
The money was put into the charge of the Königliche Gesellschaft der Wissenschaften of Göttingen, which officially announced the competition for the Wolfskehl Prize that same year:
By the power conferred on us, by Dr. Paul Wolfskehl, deceased in Darmstadt, we hereby fund a prize of one hundred thousand Marks, to be given to the person who will be the first to prove the great theorem of Fermat.
The following rules will be followed:
(1) The Königliche Gesellschaft der Wissenschaften in Göttingen will have absolute freedom to decide upon whom the prize should be conferred. It will refuse to accept any manuscript written with the sole aim of entering the competition to obtain the Prize. It will only take into consideration those mathematical memoirs which have appeared in the form of a monograph in the periodicals, or which are for sale in the bookshops. The Society asks the authors of such memoirs to send at least five printed exemplars.
(2) Works which are published in a language which is not understood by the scholarly specialists chosen for the jury will be excluded from the competition. The authors of such works will be allowed to replace them by translations, of guaranteed faithfulness.
(3) The Society declines responsibility for the examination of works not brought to its attention, as well as for the errors which might result from the fact that the author of a work, or part of a work, are unknown to the Society.
(4) The Society retains the right of decision in the case where various persons would have dealt with the solution of the problem, or for the case where the solution is the result of the combined efforts of several scholars, in particular concerning the partition of the Prize.
(5) The award of the Prize by the Society will take place not earlier than two years after the publication of the memoir to be crowned. The interval of time is intended to allow German and foreign mathematicians to voice their opinion about the validity of the solution published.
(6) As soon as the Prize is conferred by the Society, the laureate will be informed by the secretary, in the name of the Society; the result will be published wherever the Prize has been announced during the preceding year. The assignment of the Prize by the Society is not to be the subject of any further discussion.
(7) The payment of the Prize will be made to the laureate, in the next three months after the award, by the Royal Cashier of Göttingen University, or, at the receiver’s own risk, at any other place he may have designated.
(8) The capital may be delivered against receipt, at the Society’s will, either in cash, or by the transfer of financial values. The payment of the Prize will be considered as accomplished by the transmission of these financial values, even though their total value at the day’s end may not attain 100,000 Marks.
(9) If the Prize is not awarded by 13 September 2007, no ulterior claim will be accepted.
The competition for the Wolfskehl Prize is open, as of today, under the above conditions.
Göttingen, 27 June 1908
Die Königliche Gesellschaft der Wissenschaften
It is worth noti
ng that although the Committee would give 100,000 Marks to the first mathematician to prove that Fermat’s Last Theorem is true, they would not award a single pfennig to anybody who might prove that it is false.
The Wolfskehl Prize was announced in all the mathematical journals and news of the competition rapidly spread across Europe. Despite the publicity campaign and the added incentive of an enormous prize the Wolfskehl Committee failed to arouse a great deal of interest among serious mathematicians. The majority of professional mathematicians viewed Fermat’s Last Theorem as a lost cause and decided that they could not afford to waste their careers working on a fool’s errand. However, the prize did succeed in introducing the problem to a whole new audience, a hoard of eager minds who were willing to apply themselves to the ultimate riddle and approach it from a path of complete innocence.
The Era of Puzzles, Riddles and Enigmas
Ever since the Greeks, mathematicians have sought to spice up their textbooks by rephrasing proofs and theorems in the form of solutions to number puzzles. During the latter half of the nineteenth century this playful approach to the subject found its way into the popular press, and number puzzles were to be found alongside crosswords and anagrams. In due course there was a growing audience for mathematical conundrums, as amateurs contemplated everything from the most trivial riddles to profound mathematical problems, including Fermat’s Last Theorem.
