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

by Simon Singh

Life, Prince Leon, may well be compared with these public Games for in the vast crowd assembled here some are attracted by the acquisition of gain, others are led on by the hopes and ambitions of fame and glory. But among them there are a few who have come to observe and to understand all that passes here. 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.

As well as having mathematical significance for the Brotherhood, the perfection of 6 and 28 was acknowledged by other cultures who observed that the moon orbits the earth every 28 days and who declared that God created the world in 6 days.

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.

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.

can vibrate in different ways, and each vibration gives rise to a different particle.

Scientific proof is inevitably fickle and shoddy. On the other hand mathematical proof is absolute and devoid of doubt.

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.

Despite the encouragement of Father Mersenne, Fermat steadfastly refused to reveal his proofs. Publication and recognition meant nothing to him and he was satisfied with the simple pleasure of being able to create new theorems undisturbed.

The fact that he would never reveal his own proofs caused a great deal of frustration.

Although knowing π to 39 decimal places is sufficient to calculate the circumference of the universe accurate to the radius of a hydrogen atom,

During the centuries between Euclid and Diophantus, Alexandria remained the intellectual capital of the civilised world, but throughout this period the city was continually under threat from foreign armies. The first major attack occurred in 47 BC, when Julius Caesar attempted to overthrow Cleopatra by setting fire to the Alexandrian fleet. The Library, which was located near the harbour, also caught alight, and hundreds of thousands of books were destroyed.

An Arab numerologist documents the practice of carving 220 on one fruit and 284 on another, and then eating the first one and offering the second one to a lover as a form of mathematical aphrodisiac. Early theologians noted that in Genesis Jacob gave 220 goats to Esau.

Fermat discovered the pair 17,296 and 18,416.

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.

I have a truly marvellous demonstration of this proposition which this margin is too narrow to contain.

Even today it is impossible to predict the exact solution to the so-called ‘three-body problem’.

It is unthinkable for mathematicians not, in theory at least, to be able to answer every single question, and this necessity is called completeness.

The solution for Bombelli was to create a new number, i, called an imaginary number,

With a few minor manipulations imaginary numbers turn out to be the ideal way to analyse the natural swinging motion of objects such as pendula.

Number theorists consider prime numbers to be the most important numbers of all because they are the atoms of mathematics. Prime numbers are the numerical building blocks because all other numbers can be created by multiplying combinations of the prime numbers.

The question which puzzled biologists was, Why is the cicada’s life-cycle so long? And is there any significance to the life-cycle being a prime number of years? Another species, Magicicada tredecim, swarms every 13 years, implying that life-cycles lasting a prime number of years offer some evolutionary advantage.

The first woman known to have made an impact on the subject was Theano in the sixth century BC, who began as one of Pythagoras’ students before becoming one of his foremost disciples and eventually marrying him.

Hypatia, the daughter of a mathematics professor at the University of Alexandria, was famous for giving the most popular discourses in the known world and for being the greatest of problem-solvers.

Archimedes had spent his life at Syracuse, studying mathematics in relative tranquillity, but when he was in his late seventies the peace was shattered by the invading Roman army. Legend has it that during the invasion Archimedes was so engrossed in the study of a geometric figure in the sand that he failed to respond to the questioning of a Roman soldier. As a result he was speared to death.

Gauss is acknowledged as being one of the most brilliant mathematicians who has ever lived. While E.T. Bell referred to Fermat as the ‘Prince of Amateurs’, he called Gauss the ‘Prince of Mathematicians’.

‘Pure mathematicians just love a challenge. They love unsolved problems. When doing maths there’s this great feeling. You start with a problem that just mystifies you. You can’t understand it, it’s so complicated, you just can’t make head nor tail of it. But then when you finally resolve it, you have this incredible feeling of how beautiful it is, how it all fits together so elegantly. Most deceptive are the problems which look easy, and yet they turn out to be extremely intricate. Fermat is the most beautiful example of this. It just looked as though it had to have a solution and, of course, it’s very special because Fermat said that he had a solution.’

the greatest leap in calculating power since the invention of the slide-rule.

He had witnessed first-hand the turmoil caused by Gödel’s theorems of undecidability and had become involved in trying to pick up the pieces of Hilbert’s dream.

Then in 1988 Naom Elkies of Harvard University discovered the following solution:

In 1791, when he was just fourteen years old, Carl Gauss predicted the approximate manner in which the frequency of prime numbers among all the other numbers would diminish.

would underestimate the number of primes. In 1955 S. Skewes showed that the underestimate would occur sometime before reaching the number This is a number beyond the imagination, and beyond any practical application. Hardy called Skewes’s number ‘the largest number which has ever served any definite purpose in mathematics’. He calculated that if one played chess with all the particles in the universe (1087), where a move meant simply interchanging any two particles, then the number of possible games was roughly Skewes’s number. There was no reason why Fermat’s Last Theorem should not turn out to be as cruel and deceptive as Euler’s conjecture or the overestimated prime conjecture.

They got their name because in the past they were used to measure the perimeters of ellipses and the lengths of planetary orbits, but for clarity I will simply refer to them as elliptic equations rather than elliptic curves.

In the same way that biological DNA carries all the information required to construct a living organism, the E-series carries the essence of the elliptic equation.

Langlands discussed his plan for the future and tried to persuade other mathematicians to take part in what became known as the Langlands programme, a concerted effort to prove his myriad of conjectures.

‘Basically it’s just a matter of thinking. Often you write something down to clarify your thoughts, but not necessarily. In particular when you’ve reached a real impasse, when there’s a real problem that you want to overcome, then the routine kind of mathematical thinking is of no use to you. Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to really think about nothing but that problem – just concentrate on it. Then you stop. Afterwards there seems to be a kind of period of relaxation during which the subconscious appears to take over and it’s during that time that some new insight comes.’

‘I remember saying to him on a number of occasions, “It’s all very well this link to Fermat’s Last Theorem but it’s still hopeless to try and prove Taniyama–Shimura.” I think he just smiled.’

believed I could make it work, but I thought that at least I could explain why it didn’t work. I thought I was clutching at straws, but I wanted to reassure myself.