Fermat's Last Theorem

by Simon Singh

  However, the scientific theory can never be proved to the same absolute level of a mathematical theorem: it is merely considered highly likely based on the evidence available. So-called scientific proof relies on observation and perception, both of which are fallible and provide only approximations to the truth. As Bertrand Russell pointed out: ‘Although this may seem a paradox, all exact science is dominated by the idea of approximation.’ Even the most widely accepted scientific ‘proofs’ always have a small element of doubt in them. Sometimes this doubt diminishes, although it never disappears completely, while on other occasions the proof is ultimately shown to be wrong. This weakness in scientific proof leads to scientific revolutions in which one theory which was assumed to be correct is replaced with another theory, which may be merely a refinement of the original theory, or which may be a complete contradiction.

  For example, the search for the fundamental particles of matter involved each generation of physicists overturning or, at the very least, refining the theory of their predecessors. The modern quest for the building blocks of the universe started at the beginning of the nineteenth century when a series of experiments led John Dalton to suggest that everything was composed of discrete atoms, and that atoms were fundamental. At the end of the century J. J. Thomson discovered the electron, the first known subatomic particle, and therefore the atom was no longer fundamental.

  During the early years of the twentieth century, physicists developed a ‘complete’ picture of the atom – a nucleus consisting of protons and neutrons, orbited by electrons. Protons, neutrons and electrons were proudly held up as the complete ingredients for the universe. Then cosmic ray experiments revealed the existence of other fundamental particles – pions and muons. An even greater revolution came with the discovery in 1932 of antimatter – the existence of antiprotons, antineutrons, antielectrons, etc. By this time particle physicists could not be sure how many different particles existed, but at least they could be confident that these entities were indeed fundamental. That was until the 1960s when the concept of the quark was born. The proton itself is apparently built from fractionally charged quarks, as is the neutron and the pion. The moral of the story is that physicists are continually altering their picture of the universe, if not rubbing it out and starting all over again. In the next decade the very concept of a particle as a point-like object may even be replaced by the idea of particles as strings – the same strings which might best explain gravity. The theory is that strings a billionth of a billionth of a billionth of a billionth of a metre in length (so small that they appear point-like) can vibrate in different ways, and each vibration gives rise to a different particle. This is analogous to Pythagoras’ discovery that one string on a lyre can give rise to different notes depending on how it vibrates.

  The science fiction writer and futurologist Arthur C. Clarke wrote that if an eminent professor states that something is undoubtedly true, then it is likely to be proved false the next day. Scientific proof is inevitably fickle and shoddy. On the other hand mathematical proof is absolute and devoid of doubt. Pythagoras died confident in the knowledge that his theorem, which was true in 500 BC, would remain true for eternity.

  Science is operated according to the judicial system. A theory is assumed to be true if there is enough evidence to prove it ‘beyond all reasonable doubt’. On the other hand mathematics does not rely on evidence from fallible experimentation, but it is built on infallible logic. This is demonstrated by the problem of the ‘mutilated chessboard’, illustrated in Figure 2.

  Figure 2. The problem of the mutilated chessboard.

  We have a chessboard with the two opposing corners removed, so that there are only 62 squares remaining. Now we take 31 dominoes shaped such that each domino covers exactly two squares. The question is: is it possible to arrange the 31 dominoes so that they cover all the 62 squares on the chessboard?

  There are two approaches to the problem:

  (1) The scientific approach

  The scientist would try to solve the problem by experimenting, and after trying out a few dozen arrangements would discover that they all fail. Eventually the scientist believes that there is enough evidence to say that the board cannot be covered. However, the scientist can never be sure that this is truly the case because there might be some arrangement which has not been tried which might do the trick. There are millions of different arrangements and it is only possible to explore a small fraction of them. The conclusion that the task is impossible is a theory based on experiment, but the scientist will have to live with the prospect that one day the theory may be overturned.

  (2) The mathematical approach

  The mathematician tries to answer the question by developing a logical argument which will derive a conclusion which is undoubtedly correct and which will remain unchallenged forever. One such argument is the following:

  • The corners which were removed from the chessboard were both white. Therefore there are now 32 black squares and only 30 white squares.

  • Each domino covers two neighbouring squares, and neighbouring squares are always different in colour, i.e. one black and one white.

  • Therefore, no matter how they are arranged, the first 30 dominoes laid on the board must cover 30 white squares and 30 black squares.

  • Consequently, this will always leave you with one domino and two black squares remaining.

  • But remember all dominoes cover two neighbouring squares, and neighbouring squares are opposite in colour. However, the two squares remaining are the same colour and so they cannot both be covered by the one remaining domino. Therefore, covering the board is impossible!

  This proof shows that every possible arrangement of dominoes will fail to cover the mutilated chessboard. Similarly Pythagoras constructed a proof which shows that every possible right-angled triangle will obey his theorem. For Pythagoras the concept of mathematical proof was sacred, and it was proof that enabled the Brotherhood to discover so much. Most modern proofs are incredibly complicated and following the logic would be impossible for the layperson, but fortunately in the case of Pythagoras’ theorem the argument is relatively straightforward and relies on only senior school mathematics. The proof is outlined in Appendix 1.

  Pythagoras’ proof is irrefutable. It shows that his theorem holds true for every right-angled triangle in the universe. The discovery was so momentous that one hundred oxen were sacrificed as an act of gratitude to the gods. The discovery was a milestone in mathematics and one of the most important breakthroughs in the history of civilisation. Its significance was twofold. First, it developed the idea of proof. A proven mathematical result has a deeper truth than any other truth because it is the result of step-by-step logic. Although the philosopher Thales had already invented some primitive geometrical proofs, Pythagoras took the idea much further and was able to prove far more ingenious mathematical statements. The second consequence of Pythagoras’ theorem is that it ties the abstract mathematical method to something tangible. Pythagoras showed that the truth of mathematics could be applied to the scientific world and provide it with a logical foundation. Mathematics gives science a rigorous beginning and upon this infallible foundation scientists add inaccurate measurements and imperfect observations.

  An Infinity of Triples

  The Pythagorean Brotherhood invigorated mathematics with its zealous search for truth via proof. News of their success spread and yet the details of their discoveries remained a closely guarded secret. Many requested admission to the inner sanctum of knowledge, but only the most brilliant minds were accepted. One of those who was blackballed was a candidate by the name of Cylon. Cylon took exception to his humiliating rejection and twenty years later he took his revenge.

  During the sixty-seventh Olympiad (510 BC) there was a revolt in the nearby city of Sybaris. Telys, the victorious leader of the revolt, began a barbaric campaign of persecution against the supporters of the former government, which drove many of them to seek sanctuary in Croton. Telys
demanded that the traitors be returned to Sybaris to suffer their due punishment, but Milo and Pythagoras persuaded the citizens of Croton to stand up to the tyrant and protect the refugees. Telys was furious and immediately gathered an army of 300,000 men and marched on Croton, where Milo defended the city with 100,000 armed citizens. After seventy days of war Milo’s supreme generalship led him to victory and as an act of retribution he turned the course of the river Crathis upon Sybaris to flood and destroy the city.

  Despite the end of the war, the city of Croton was still in turmoil because of arguments over what should be done with the spoils of war. Fearful that the lands would be given to the Pythagorean elite, the ordinary folk of Croton began to grumble. There had already been growing resentment among the masses because the secretive Brotherhood continued to withold their discoveries, but nothing came of it until Cylon emerged as the voice of the people. Cylon preyed on the fear, paranoia and envy of the mob and led them on a mission to destroy the most brilliant school of mathematics the world had ever seen. Milo’s house and the adjoining school were surrounded, all the doors were locked and barred to prevent escape and then the burning began. Milo fought his way out of the inferno and fled, but Pythagoras, along with many of his disciples, was killed.

  Mathematics had lost its first great hero, but the Pythagorean spirit lived on. The numbers and their truths were immortal. Pythagoras had demonstrated that more than any other discipline mathematics is a subject which is not subjective. His disciples did not need their master to decide on the validity of a particular theory. A theory’s truth was independent of opinion. Instead the construction of mathematical logic had become the arbiter of truth. This was the Pythagoreans’ greatest contribution to civilisation – a way of achieving truth which is beyond the fallibility of human judgement.

  Following the death of their founder and the attack by Cylon, the Brotherhood left Croton for other cities in Magna Graecia, but the persecution continued and eventually many of them had to settle in foreign lands. This enforced migration encouraged the Pythagoreans to spread their mathematical gospel throughout the ancient world. Pythagoras’ disciples set up new schools and taught their students the method of logical proof. In addition to their proof of Pythagoras’ theorem, they also explained to the world the secret of finding so-called Pythagorean triples.

  Figure 3. Finding whole number solutions to Pythagoras’ equation can be thought of in terms of finding two squares which can be added together to form a third square. For example, a square made of 9 tiles can be added to a square of 16 tiles, and rearranged to form a third square made of 25 tiles.

  Pythagorean triples are combinations of three whole numbers which perfectly fit Pythagoras’ equation: x2 + y2 = z2 For example, Pythagoras’ equation holds true if x = 3, y = 4 and z = 5:

  Another way to think of Pythagorean triples is in terms of rearranging squares. If one has a 3 × 3 square made of 9 tiles, and a 4 × 4 square made of 16 tiles, then all the tiles can be rearranged to form a 5 × 5 square made of 25 tiles, as shown in Figure 3.

  The Pythagoreans wanted to find other Pythagorean triples, other squares which could be added to form a third, larger square. Another Pythagorean triple is x = 5, y = 12 and z = 13:

  A larger Pythagorean triple is x = 99, y = 4,900 and z = 4,901. Pythagorean triples become rarer as the numbers increase, and finding them becomes harder and harder. To discover as many triples as possible the Pythgoreans invented a methodical way of finding them, and in so doing they also demonstrated that there are an infinite number of Pythagorean triples.

  From Pythagoras’ Theorem to Fermat’s Last Theorem

  Pythagoras’ theorem and its infinity of triples was discussed in E.T. Bell’s The Last Problem, the library book which caught the attention of the young Andrew Wiles. Although the Brotherhood had achieved an almost complete understanding of Pythagorean triples, Wiles soon discovered that this apparently innocent equation, x2 + y2 = z2, has a darker side – Bell’s book described the existence of a mathematical monster.

  In Pythagoras’ equation the three numbers, x, y and z, are all squared (i.e. x2 = x × x):

  However, the book described a sister equation in which x, y and z are all cubed (i.e. x3 = x × x × x). The so-called power of x in this equation is no longer 2, but rather 3:

  Finding whole number solutions, i.e. Pythagorean triples, to the original equation was relatively easy, but changing the power from ‘2’ to ‘3’ (the square to a cube) and finding whole number solutions to the sister equation appears to be impossible. Generations of mathematicians scribbling on notepads have failed to find numbers which fit the equation perfectly.

  Figure 4. Is it possible to add the building blogs from one cube to another cube, to form a third, larger cube? In this case a 6 × 6 × 6 cube added to an 8 × 8 × 8 cube does not have quite enough building blocks to form a 9 × 9 × 9 cube. There are 216 (63) building blocks in the first cube, and 512 (83) in the second. The total is 728 building blogs, which is 1 short of 93.

  With the original ‘squared’ equation, the challenge was to rearrange the tiles in two squares to form a third, larger square. The ‘cubed’ version of the challenge is to rearrange two cubes made of building blocks, to form a third, larger cube. Apparently, no matter what cubes are chosen to begin with, when they are combined the result is either a complete cube with some extra blocks left over, or an incomplete cube. The nearest that anyone has come to a perfect rearrangement is one in which there is one building block too many or too few. For example, if we begin with the cubes 63 (x3) and 83 (y3) and rearrange the building blocks, then we are only one short of making a complete 9 × 9 × 9 cube, as shown in Figure 4.

  Finding three numbers which fit the cubed equation perfectly seems to be impossible. That is to say, there appear to be no whole number solutions to the equation

  Furthermore, if the power is changed from 3 (cubed) to any higher number n (i.e. 4, 5, 6, …), then finding a solution still seems to be impossible. There appear to be no whole number solutions to the more general equation

  By merely changing the 2 in Pythagoras’ equation to any higher number, finding whole number solutions turns from being relatively simple to being mind-bogglingly difficult. In fact, the great seventeenth-century Frenchman Pierre de Fermat made the astonishing claim that the reason why nobody could find any solutions was that no solutions existed.

  Fermat was one of the most brilliant and intriguing mathematicians in history. He could not have checked the infinity of numbers, but he was absolutely sure that no combination existed which would fit the equation perfectly because his claim was based on proof. Like Pythagoras, who did not have to check every triangle to demonstrate the validity of his theorem, Fermat did not have to check every number to show the validity of his theorem. Fermat’s Last Theorem, as it is known, stated that

  has no whole number solutions for n greater than 2.

  As Wiles read each chapter of Bell’s book, he learnt how Fermat had become fascinated by Pythagoras’ work and had eventually come to study the perverted form of Pythagoras’ equation. He then read how Fermat had claimed that even if all the mathematicians in the world spent eternity looking for a solution to the equation they would fail to find one. He must have eagerly turned the pages, relishing the thought of examining the proof of Fermat’s Last Theorem. However, the proof was not there. It was not anywhere. Bell ended the book by stating that the proof had been lost long ago. There was no hint of what it might have been, no clues as to the proof’s construction or derivation. Wiles found himself puzzled, infuriated and intrigued. He was in good company.

  For over 300 years many of the greatest mathematicians had tried to rediscover Fermat’s lost proof and failed. As each generation failed, the next became even more frustrated and determined. In 1742, almost a century after Fermat’s death, the Swiss mathematician Leonhard Euler asked his friend Clêrot to search Fermat’s house in case some vital scrap of paper still remained. No clues were ever found as to w
hat Fermat’s proof might have been. In Chapter 2 we shall find out more about the mysterious Pierre de Fermat and how his theorem came to be lost, but for the time being it is enough to know that Fermat’s Last Theorem, a problem that had captivated mathematicians for centuries, had captured the imagination of the young Andrew Wiles.

  Sat in Milton Road Library was a ten-year-old boy staring at the most infamous problem in mathematics. Usually half the difficulty in a mathematics problem is understanding the question, but in this case it was simple – prove that xn + yn = zn has no whole number solutions for n greater than 2. Andrew was not daunted by the knowledge that the most brilliant minds on the planet had failed to rediscover the proof. He immediately set to work using all his textbook techniques to try and recreate the proof. Perhaps he could find something that everyone else, except Fermat, had overlooked. He dreamed he could shock the world.

  Thirty years later Andrew Wiles was ready. Standing in the auditorium of the Isaac Newton Institute, he scribbled on the board and then, struggling to contain his glee, stared at his audience. The lecture was reaching its climax and the audience knew it. One or two of them had smuggled cameras into the lecture room and flashes peppered his concluding remarks.

  With the chalk in his hand he turned to the board for the last time. The final few lines of logic completed the proof. For the first time in over three centuries Fermat’s challenge had been met. A few more cameras flashed to capture the historic moment. Wiles wrote up the statement of Fermat’s Last Theorem, turned towards the audience, and said modestly: ‘I think I’ll stop here.’