The fame of Fermat’s Last Theorem comes solely from the sheer difficulty of proving it. An extra sparkle is added by the fact that the Prince of Amateurs said that he could prove this theorem which has since baffled generations of professional mathematicians. Fermat’s offhand comments in the margin of his copy of the Arithmetica were read as a challenge to the world. He had proved the Last Theorem: the question was, could any mathematician match his brilliance?
G.H. Hardy had a whimsical sense of humour and dreamt up what could have been an equally frustrating legacy. Hardy’s challenge was in the form of an insurance policy to help him cope with his fear of travelling on ships. If he ever had to journey across the sea he would first send a telegram to a colleague saying:
HAVE SOLVED RIEMANN HYPOTHESIS STOP
WILL GIVE DETAILS UPON RETURN STOP
The Riemann hypothesis is a problem which has plagued mathematicians since the nineteenth century. Hardy’s logic was that God would never allow him to drown because it would leave mathematicians haunted by a second terrible phantom.
Fermat’s Last Theorem is a problem of immense difficulty, and yet it can be stated in a form that a schoolchild can understand. There can be no problem in physics, chemistry or biology which can be so simply and unambiguously stated and which has remained unsolved for so long. In his book The Last Problem, E.T. Bell wrote that civilisation would probably come to an end before Fermat’s Last Theorem could be solved. Proving Fermat’s Last Theorem has become the most valuable prize in number theory, and not surprisingly it has led to some of the most exciting episodes in the history of mathematics. The search for a proof of Fermat’s Last Theorem has involved the greatest minds on the planet, huge rewards, suicidal despair and duelling at dawn.
The riddle’s status has gone beyond the closed world of mathematics. In 1958 it even made its way into a Faustian tale. An anthology entitled Deals with the Devil contains a short story written by Arthur Porges. In ‘The Devil and Simon Flagg’ the Devil asks Simon Flagg to set him a question. If the Devil succeeds in answering it within twenty-four hours then he takes Simon’s soul, but if he fails then he must give Simon $100,000. Simon poses the question: ‘Is Fermat’s Last Theorem correct?’ The Devil disappears and whizzes around the world to absorb every piece of mathematics that has ever been created. The following day he returns and admits defeat:
‘You win, Simon,’ he said, almost in a whisper, eyeing him with ungrudging respect. ‘Not even I can learn enough mathematics in such a short time for so difficult a problem. The more I got into it the worse it became. Non-unique factoring, ideals – Bah! Do you know,’ the Devil confided, ‘not even the best mathematicians on other planets – all far ahead of yours – have solved it? Why, there’s a chap on Saturn – he looks something like a mushroom on stilts – who solves partial differential equations mentally; and even he’s given up.’
3
A Mathematical Disgrace
Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost. Rigour should be a signal to the historian that the maps have been made, and the real explorers have gone elsewhere.
W.S. Anglin
‘Since I first met Fermat’s Last Theorem as a child it’s been my greatest passion,’ recalls Andrew Wiles, in a hesitant voice which conveys the emotion he feels about the problem. ‘I’d found this problem which had been unsolved for three hundred years. I don’t think many of my schoolfriends caught the mathematics bug, so I didn’t discuss it with my contemporaries. But I did have a teacher who had done research in mathematics and he gave me a book about number theory that gave me some clues about how to start tackling it. To begin with I worked on the assumption that Fermat didn’t know very much more mathematics than I would have known. I tried to find his lost solution by using the kind of methods he might have used.’
Wiles was a child full of innocence and ambition, who saw an opportunity to succeed where generations of mathematicians had failed. To others this might have seemed like a foolhardy dream but young Andrew was right in thinking that he, a twentieth-century schoolboy, knew as much mathematics as Pierre de Fermat, a genius of the seventeenth century. Perhaps in his naïvety he would stumble upon a proof which other more sophisticated minds had missed.
Despite his enthusiasm every calculation resulted in a dead end. Having racked his brains and sifted through his schoolbooks he was achieving nothing. After a year of failure he changed his strategy and decided that he might be able to learn something from the mistakes of other more eminent mathematicians. ‘Fermat’s Last Theorem has this incredible romantic history to it. Many people have thought about it, and the more that great mathematicians in the past have tried and failed to solve the problem, the more of a challenge and the more of a mystery it’s become. Many mathematicians had tried it in so many different ways in the eighteenth and nineteenth centuries, and so as a teenager I decided that I ought to study those methods and try to understand what they’d been doing.’
Young Wiles examined the approaches of everyone who had ever made a serious attempt to prove Fermat’s Last Theorem. He began by studying the work of the most prolific mathematician in history and the first one to make a breakthrough in the battle against Fermat.
The Mathematical Cyclops
Creating mathematics is a painful and mysterious experience. Often the object of the proof is clear, but the route is shrouded in fog, and the mathematician stumbles through a calculation, terrified that each step might be taking the argument in completely the wrong direction. Additionally there is the fear that no route exists. A mathematician may believe that a statement is true, and spend years trying to prove that it is indeed true, when all along it is actually false. The mathematician has effectively been attempting to prove the impossible.
In the entire history of the subject only a handful of mathematicians appear to have avoided the self-doubt which intimidates their colleagues. Perhaps the most notable example of such a mathematician was the eighteenth-century genius Leonhard Euler, and it was he who made the first breakthrough towards proving Fermat’s Last Theorem. Euler had such an incredible intuition and vast memory that it was said he could map out the entire bulk of a calculation in his head without having to put pen to paper. Across Europe he was referred to as ‘analysis incarnate’, and the French academician François Arago said, ‘Euler calculated without apparent effort as men breathe, or as eagles sustain themselves in the wind.’
Leonhard Euler was born in Basle in 1707, the son of a Calvinist pastor, Paul Euler. Although the young Euler showed a prodigious talent for mathematics, his father was determined that he should study theology and pursue a career in the Church. Leonhard dutifully obeyed and studied theology and Hebrew at the University of Basle.
Fortunately for Euler the town of Basle was also home to the eminent Bernoulli clan. The Bernoullis could easily claim to be the most mathematical of families, creating eight of Europe’s most outstanding minds within only three generations – some have said that the Bernoulli family was to mathematics what the Bach family was to music. Their fame spread beyond the mathematical community and one particular legend typifies the profile of the family. Daniel Bernoulli was once travelling across Europe and had struck up a conversation with a stranger. After a while he modestly introduced himself: ‘I am Daniel Bernoulli.’ ‘And I,’ said his companion sarcastically, ‘am Isaac Newton.’ Daniel fondly recalled this incident on several occasions, considering it the most sincere tribute he had ever received.
Daniel and Nikolaus Bernoulli were close friends of Leonhard Euler, and they realised that the most brilliant of mathematicians was being turned into the most mediocre of theologians. They appealed to Paul Euler and requested that Leonhard be allowed to forsake the cloth in favour of numbers. Euler senior had in the past been taught mathematics by Bernoulli senior, Jakob, and had a tremendous respect for the family. Reluctantly he accepted that his son had been born to cal
culate, not preach.
Leonhard Euler soon left Switzerland for the palaces of Berlin and St Petersburg, where he was to spend the bulk of his creative years. During the era of Fermat, mathematicians were considered amateur number-jugglers, but by the eighteenth century they were treated as professional problem-solvers. The culture of numbers had changed dramatically, and this was partly a consequence of Sir Isaac Newton and his scientific calculations.
Newton believed that mathematicians were wasting their time teasing each other with pointless riddles. Instead he would apply mathematics to the physical world and calculate everything from the orbits of the planets to the trajectories of cannon-balls. By the time Newton died, in 1727, Europe had undergone a scientific revolution, and in the same year Euler published his first paper. Although the paper contained elegant and innovative mathematics, it was primarily aimed at describing a solution to a technical problem regarding the masting of ships.
The European powers were not interested in using mathematics to explore esoteric and abstract concepts; instead they wanted to exploit mathematics to solve practical problems, and they competed to employ the best minds. Euler began his career with the Czars, before being invited to the Berlin Academy by Frederick the Great of Prussia. Eventually he returned to Russia, under the rule of Catherine the Great, where he spent his final years. During his career he tackled a multitude of problems, ranging from navigation to finance, and from acoustics to irrigation. The practical world of problem-solving did not dull Euler’s mathematical ability. Instead tackling each new task would inspire him to create innovative and ingenious mathematics. His single-minded passion drove him to write several papers in a single day, and it is said that between the first and second calls for dinner he would attempt to dash off a complete calculation worthy of publication. Not a moment was wasted and even when he was cradling an infant in one hand Euler would be outlining a proof with the other.
One of Euler’s greatest achievements was the development of the algorithmic method. The point of Euler’s algorithms was to tackle apparently impossible problems. One such problem was predicting the phases of the moon far into the future with high accuracy – information which could be used to draw up vital navigation tables. Newton had already shown that it is relatively easy to predict the orbit of one body around another, but in the case of the moon the situation is not so simple. The moon orbits the earth, but there is a third body, the sun, which complicates matters enormously. While the earth and moon attract each other, the sun perturbs the position of the earth and has a knock-on effect on the orbit of the moon. Equations could be used to pin down the effect of any two of the bodies, but eighteenth-century mathematicians could not incorporate the third body into their calculations. Even today it is impossible to predict the exact solution to the so-called ‘three-body problem’.
Euler realised that mariners did not need to know the phase of the moon with absolute accuracy, only with enough precision to locate their own position to within a few nautical miles. Consequently Euler developed a recipe for generating an imperfect but sufficiently accurate solution. The recipe, known as an algorithm, worked by first obtaining a rough-and-ready result, which could then be fed back into the algorithm to generate a more refined result. This refined result could then be fed back into the algorithm to generate an even more accurate result, and so on. A hundred or so iterations later Euler was able to provide a position for the moon which was accurate enough for the purposes of the navy. He gave his algorithm to the British Admiralty and in return they rewarded him with a prize of £300.
Euler earned a reputation for being able to solve any problem which was posed, a talent which seemed to extend even beyond the realm of science. During his stint at the court of Catherine the Great he encountered the great French philosopher Denis Diderot. Diderot was a committed atheist and would spend his days converting the Russians to atheism. This infuriated Catherine, who asked Euler to put a stop to the efforts of the godless Frenchman.
Euler gave the matter some thought and claimed that he had an algebraic proof for the existence of God. Catherine the Great invited Euler and Diderot to the palace and gathered together her courtiers to listen to the theological debate. Euler stood before the audience and announced:
With no great understanding of algebra, Diderot was unable to argue against the greatest mathematician in Europe and was left speechless. Humiliated, he left St Petersburg and returned to Paris. In his absence, Euler continued to enjoy his return to theological study and published several other mock proofs concerning the nature of God and the human spirit.
Figure 5. The River Pregel divides the town of Königsberg into four seperate parts, A, B, C and D. Seven bridges connect the various parts of the town, and a local riddle asked if it was possible to make a journey such that each bridge is crossed once and only once.
A more valid problem which also appealed to Euler’s whimsical nature concerned the Prussian city of Königsberg, now known as the Russian city of Kaliningrad. The city is built on the banks of the river Pregel and consists of four separate quarters connected by seven bridges. Figure 5 shows the layout of the city. Some of the more curious residents of Königsberg wondered if it was possible to plot a journey across all seven bridges without having to stroll across any bridge more than once. The citizens of Königsberg tried various routes but each one ended in failure. Euler also failed to find a successful route, but he was successful in explaining why making such a journey was impossible.
Figure 6. A simplified representation of the bridges of Königsberg
Euler began with a plan of the city, and from it he generated a simplified representation in which the sections of land were reduced to points and bridges were replaced by lines, as shown in Figure 6. He then argued that, in general, in order to make a successful journey (i.e. crossing all bridges only once) a point should be connected to an even number of lines. This is because in the middle of a journey when the traveller passes through a land mass, he or she must enter via one bridge and then leave via a different bridge. There are only two exceptions to this rule – when a traveller either begins or ends the journey. At the start of the journey the traveller leaves a land mass and requires only a single bridge to exit, and at the end of the journey the traveller arrives at a land mass and requires only a single bridge to enter. If the journey begins and ends in different locations, then these two land masses are allowed to have an odd number of bridges. But if the journey begins and ends in the same place, then this point, like all the other points, must have an even number of bridges.
So, in general, Euler concluded that, for any network of bridges, it is only possible to make a complete journey crossing each bridge only once if all the landmasses have an even number of bridges, or exactly two land masses have an odd number of bridges. In the case of Königsberg there are four land masses in total and all of them are connected to an odd number of bridges – three points have three bridges, and one has five bridges. Euler had been able to explain why it was impossible to cross each one of Königsberg’s bridges once and only once, and furthermore he had generated a rule which could be applied to any network of bridges in any city in the world. The argument is beautifully simple, and was perhaps just the sort of logical problem that Euler dashed off before dinner.
The Königsberg bridge puzzle is a so-called network problem in applied mathematics, but it inspired Euler to consider more abstract networks. He went on to discover a fundamental truth about all networks, the so-called network formula, which he could prove with just a handful of logical steps. The network formula shows an eternal relationship between the three properties which describe any network:
where
V = the number of vertices (intersections) in the network,
L = the number of lines in the network,
R = the number of regions (enclosed areas) in the network.
Euler claimed that for any network one could add the number of vertices and regions and subtract the number of lines and
the total would always be 1. For example, all the networks in Figure 7 obey the rule.
Figure 7. All conceivable networks obey Euler’s network formula.
It is possible to imagine testing this formula on a whole series of networks and if it turned out to be true on each occasion it would be tempting to assume that the formula is true for all networks. Although this might be enough evidence for a scientific theory, it is inadequate to justify a mathematical theorem. The only way to show that the formula works for every possible network is to construct a foolproof argument, which is exactly what Euler did.
Euler began by considering the simplest network of all, i.e. a single vertex as shown in Figure 8. For this network the formula is clearly true: there is one vertex, and no lines or regions, and therefore
Euler then considered what would happen if he added something to this simplest of all networks. Any extension to the single vertex requires the addition of a line. The line can either connect the existing vertex to itself, or it can connect the existing vertex to a new vertex.