Fermat’s Last Theorem: The compelling biography and history of mathematical intellectual endeavour

by Simon Singh

Proof is what lies at the heart of maths, and is what marks it out from other sciences. Other sciences have hypotheses that are tested against experimental evidence until they fail, and are overtaken by new hypotheses. In maths, absolute proof is the goal, and once something is proved, it is proved forever, with no room for change.

The film was transmitted on BBC Television as Horizon: Fermat’s Last Theorem.

In mathematics the experience that comes with age seems less important than the intuition and daring of youth.

The word itself, geometry, means ‘to measure the earth’.

It is the same with life. Some are influenced by the love of wealth while others are blindly led on by the mad fever for power and domination, but the finest type of man gives himself up to discovering the meaning and purpose of life itself. He seeks to uncover the secrets of nature. This is the man I call a philosopher for although no man is completely wise in all respects, he can love wisdom as the key to nature’s secrets.

The most significant and rarest numbers are those whose divisors add up exactly to the number itself and these are the perfect numbers. The number 6 has the divisors 1, 2 and 3, and consequently it is a perfect number because 1 + 2 + 3 = 6. The next perfect number is 28, because 1 + 2 + 4 + 7 + 14 = 28.

The third perfect number is 496, the fourth is 8,128, the fifth is 33,550,336 and the sixth is 8,589,869,056.

perfect numbers are always the sum of a series of consecutive counting numbers. So we have

numbers, one of which is a power of 2 and the other being the next power of 2 minus 1. That is to say,

He analysed the hammers and realised that those which were harmonious with each other had a simple mathematical relationship – their masses were simple ratios or fractions of each other. That is to say that hammers half, two-thirds or three-quarters the weight of a particular hammer would all generate harmonious sounds.

For example, one particular number appears to guide the lengths of meandering rivers. Professor Hans-Henrik Stølum, an earth scientist at Cambridge University, has calculated the ratio between the actual length of rivers from source to mouth and their direct length as the crow flies. Although the ratio varies from river to river, the average value is slightly greater than 3, that is to say that the actual length is roughly three times greater than the direct distance. In fact the ratio is approximately 3.14, which is close to the value of the number π, the ratio between the circumference of a circle and its diameter.

this point it is important to note that, although this theorem will forever be associated with Pythagoras, it was actually used by the Chinese and the Babylonians one thousand years before. However, these cultures did not know that the theorem was true for every right-angled triangle. It was certainly true for the triangles they tested, but they had no way of showing that it was true for all the right-angled triangles which they had not tested. The reason for Pythagoras’ claim to the theorem is that it was he who first demonstrated its universal truth.

The idea of a classic mathematical proof is to begin with a series of axioms, statements which can be assumed to be true or which are self-evidently true. Then by arguing logically, step by step, it is possible to arrive at a conclusion. If the axioms are correct and the logic is flawless, then the conclusion will be undeniable. This conclusion is the theorem.

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.

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

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 mathema...

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.

Figure 4. Is it possible to add the building blocks 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 blocks, which is 1 short of 93.

Fermat’s Last Theorem, as it is known, stated that has no whole number solutions for n greater than 2.

This exchange of letters with Pascal,

concerned the creation of an entirely new branch of mathematics – probability theory.

One of the most counterintuitive probability problems concerns the likelihood of sharing birthdays. Imagine a football pitch with 23 people on it, the players and the referee. What is the probability that any two of those 23 people share the same birthday? With 23 people and 365 birthdays to chose from, it would seem highly unlikely that anybody would share the same birthday. If asked to put a figure on it most people would guess a probability of perhaps 10% at most. In fact, the actual answer is just over 50% – that is to say, on the balance of probability, it is more likely than not that two people on the pitch will share the same birthday. The reason for this high probability is that what matters more than the number of people is the number of ways people can be paired. When we look for a shared birthday, we need to look at pairs of people not individuals. Whereas there are only 23 people on the pitch, there are 253 pairs of people. For example, the first person can be paired with any of the other 22 people giving 22 pairings to start with. Then, the second person can be paired with any of the remaining 21 people (we have already counted the second person paired with the first person so the number of possible pairings is reduced by one), giving an additional 21 pairings. Then, the third person can be paired with any of the remaining 20 people, giving an additional 20 pairings, and so on until we reach a total of 253 pairs.

Pascal was even convinced that he could use his theories to justify a belief in God. He stated that ‘the excitement that a gambler feels when making a bet is equal to the amount he might win multiplied by the probability of winning it’. He then argued that the possible prize of eternal happiness has an infinite value and that the probability of entering heaven by leading a virtuous life, no matter how small, is certainly finite. Therefore, according to Pascal’s definition, religion was a game of infinite excitement and one worth playing, because multiplying an infinite prize by a finite probability results in infinity.

Calculus is the ability to calculate the rate of change, known as the derivative, of one quantity with respect to another.

Therefore √2 cannot be written as a fraction and is irrational. An outline of Euclid’s proof is given in Appendix 2. By using proof by contradiction Euclid was able to prove the existence of irrational numbers. For the first time numbers had taken on a new and more abstract quality. Until this point in history all numbers could be expressed as whole numbers or fractions, but Euclid’s irrational numbers defied representation in the traditional manner. There is no other way to describe the number equal to the square root of two other than by expressing it as √2, because it cannot be written as a fraction and any attempt to write it as a decimal could only ever be an approximation, e.g. 1.414213562373 …

The challenge is to calculate Diophantus’ life span. The answer can be found in Appendix 3. This riddle is an example of the sort of problem that Diophantus relished. His speciality was to tackle questions which required whole number solutions, and today such questions are referred to as Diophantine problems. He spent his career in Alexandria collecting well-understood problems and inventing new ones, and then compiled them all into a major treatise entitled Arithmetica. Of the thirteen books which made up the Arithmetica, only six would survive the turmoils of the Dark Ages and go on to inspire the Renaissance mathematicians, including Pierre de Fermat. The remaining seven books would be lost during a series of tragic events which would send mathematics back to the age of the Babylonians.

Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number. The Pythagoreans made the extraordinary discovery that 220 and 284 are friendly numbers. The divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, and the sum of all these is 284. On the other hand the divisors of 284 are 1, 2, 4, 71, 142, and the sum of all these is 220.

Although discovering a new pair of friendly numbers made Fermat something of a celebrity, his reputation was truly confirmed thanks to a series of mathematical challenges. For example, Fermat noticed that 26 is sandwiched between 25 and 27, one of which is a square number (25 = 52 = 5 × 5) and the other is a cube number (27 = 33 = 3 × 3 × 3). He searched for other numbers sandwiched between a square and a cube but failed to find any, and suspected that 26 might be unique. After days of strenuous effort he managed to construct an elaborate argument which proved without any doubt that 26 is indeed the only number between a square and a cube. His step-by-step logical proof established that no other numbers could fulfil this criterion. Fermat announced this unique property of 26 to the mathematical community, and then challenged them to prove that this was the case. He openly admitted that he himself had a proof; the question was, however, did others have the ingenuity to match it?

According to Fermat there appeared to be no three numbers which would perfectly fit the equation

In fact for the last 350 years Fermat’s Last Theorem should more accurately have been referred to as Fermat’s Last Conjecture. As the centuries passed, all his other observations were proved one by one, but Fermat’s Last Theorem stubbornly refused to give in so easily. In fact, it is called the ‘Last’ Theorem because it remains the last one of the observations to be proved.

this acknowledged difficulty does not necessarily mean that Fermat’s Last Theorem is an important theorem in the ways described earlier. The Last Theorem, at least until very recently, seemed to fail to fulfil several criteria – it seemed that proving it would not lead to anything profound, it would not give any particularly deep insight about numbers, and it would not help prove any other conjectures.

The fame of Fermat’s Last Theorem comes solely from the sheer difficulty of proving it.

The practical world of problem-solving did not dull Euler’s mathematical

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.

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.

However, if you examine the arrangement which was being sold by Loyd, in which the 14 and 15 were swapped, then the value of the disorder parameter is one, Dp = 1, i.e. the only pair of tiles out of order are the 14 and 15. For Loyd’s arrangement the disorder parameter has an odd value! Yet we know that any arrangement derived from the correct arrangement has an even value for the disorder parameter. The conclusion is that Loyd’s arrangement cannot be derived from the correct arrangement, and conversely it is impossible to get from Loyd’s arrangement back to the correct one – Loyd’s $1,000 was safe.

the great number theorist Andre Weil would say: ‘God exists since mathematics is consistent, and the Devil exists since we cannot prove it.’

The Missing Link

By a deft series of complicated manoeuvres Frey fashioned Fermat’s original equation, with the hypothetical solution, into

Elliptic equations have the form but if we let

Therefore the existence of Frey’s elliptic equation defies the Taniyama–Shimura conjecture. In other words, Frey’s argument was as follows: (1) If (and only if) Fermat’s Last Theorem is wrong, then Frey’s elliptic equation exists. (2) Frey’s elliptic equation is so weird that it can never be modular. (3) The Taniyama–Shimura conjecture claims that every elliptic equation must be modular. (4) Therefore the Taniyama–Shimura conjecture must be false! Alternatively, and more importantly, Frey could run his argument backwards: (1) If the Taniyama–Shimura conjecture can be proved to be true, then every elliptic equation must be modular. (2) If every elliptic equation must be modular, then the Frey elliptic equation is forbidden to exist. (3) If the Frey elliptic equation does not exist, then there can be no solutions to Fermat’s equation. (4) Therefore Fermat’s Last Theorem is true!

he could only prove a very minor part of it. ‘I sat down with Barry and told him what I was working on. I mentioned that I’d proved a very special case, but I didn’t know what to do next to generalise it to get the full strength of the proof.’ Professor Mazur sipped his cappuccino and listened to Ribet’s idea. Then he stopped and stared at Ken in disbelief. ‘But don’t you see? You’ve already done it! All you have to do is add some gamma-zero of (M) structure and just run through your argument and it works. It gives you everything you need.’

Instead of using the infinite groups, Galois began with a particular equation and constructed his group from the handful of solutions to that equation. It was groups formed from the solutions to quintic equations which allowed Galois to derive his results about these equations. A century and a half later Wiles would use Galois’s work as the foundation for his proof of the Taniyama–Shimura conjecture.

In hindsight this seemed like the natural route to a proof, but getting this far had required enormous determination to overcome the periods of self-doubt. Wiles describes his experience of doing mathematics in terms of a journey through a dark unexplored mansion. ‘One enters the first room of the mansion and it’s dark. Completely dark. One stumbles around bumping into the furniture but gradually you learn where each piece of furniture is. Finally, after six months or so, you find the light switch, you turn it on, and suddenly it’s all illuminated. You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them.’

The Lecture of the Century After seven years of single-minded effort Wiles had completed a proof of the Taniyama–Shimura conjecture. As a consequence, and after thirty years of dreaming about it, he had also proved Fermat’s Last Theorem. It was now time to tell the rest of the world.

the 200-page proof

Appendix 4. Bachet’s Weighing Problem In order to weigh any whole number of kilograms from 1 to 40 most people will suggest that six weights are required: 1, 2, 4, 8, 16, 32 kg. In this way, all the weights can easily be achieved by placing the following combinations in one pan: However, by placing weights in both pans, such that weights are also allowed to sit alongside the object being weighed, Bachet could complete the task with only four weights: 1, 3, 9, 27 kg. A weight placed in the same pan as the object being weighed effectively assumes a negative value. Thus, the weights can be achieved as follows: