Now this is extraordinary because with an infinity of numbers to choose from, you would expect to find some that would fit the equation. But Fermat was sure that this was impossible - because he had a proof. He wrote in the margin of the book (the Arithmetica), that the equation had no solutions, and then he wrote that he had a ‘demonstratum mirabelis’, a marvellous proof, and then he wrote that he did not have enough space on the page to write down that marvellous proof.

And that’s what makes Fermat’s Last Theorem so wonderful. He said he had a proof, but he never wrote it down. After his death, people read his scribbled notes and thus began a search to rediscover Fermat’s proof. Mathematicians devoted entire careers to the Last Theorem.

But if proving Fermat’s Last Theorem is only a riddle, of no practical value, then why did mathematicians bother trying to prove it?

 Well, that’s what mathematicians do. The joy of mathematics is solving a problem just because it’s there. Each problem is an intellectual Everest. And the harder the problem is, the greater the potential joy of solving it. Fermat actually made lots of infuriating notes about proving different theorems and after about a century they had all been re-proved except for the Last Theorem - that’s why it’s called the Last Theorem. And because it was the Last Theorem it was the hardest one to prove, and therefore it became even more desirable to prove it. It became the most famous problem in mathematics.
The greatest mathematician of the 18th century was Leonhard Euler - he tried to prove the Last Theorem and failed. After that, anyone who proved it could claim to be greater than Euler.

But Fermat's Last Theorem looks beautifully simple. Is it really so hard to prove?

One mathematician told me that Fermat's Last Theorem is a 'blue sky question'. By this he meant that anybody can ask, why is the sky blue? This is a very simply question, a child can ask it. However, answering this question is very difficult, because it requires an understanding of the physics of light and scattering, it requires an understanding of dust particles and air at the molecular level, and it requires an understanding of quantum physics. Fermat's Last Theorem is the same - an easy question to ask, a very difficult one to answer.

The problem is so appealing to mathematicians because of its simplicity, elegance and difficulty. The longer that  mathematicians failed to answer the question, the more they wanted to answer it. And, of course, Pierre de Fermat in the seventeenth century said that he had an answer, which meant that it was even more frustrating that modern mathematicians did not have an answer.

So generations of mathematicians began to tackle Fermat’s challenge, and one of the most significant breakthroughs was by Sophie Germain - who was she?

Sophie Germain was born at the start of the 19th century, when mathematics was considered unsuitable for women. Her parents tried to discourage her by confiscating her clothes and candles so she could not study in the evenings, but she carried on regardless and eventually her parents accepted her passion. When she was older she could not attend the Ecole Polytechnique, because this was a men-only college. So she took on the identity of a man, Monsieur Leblanc, in order to study at the Polytechnique. Eventually she established her reputation as a mathematician and could reveal her true identity. She made the biggest breakthrough during the 19th century towards proving the Last Theorem, but unfortunately she could only prove a part of it, not all of it. People tend to think of mathematicians as dry, dull people but the history of the Last Theorem is full of obsessed, passionate characters, such as Sophie German.

Then there’s Paul Wolfskehl - what was his contribution to proving the Last Theorem?

Wolfskehl was a German industrialist and amateur mathematician at the end of the last century. He had been rejected by his true love, and had decided to commit suicide at midnight. While waiting for the appointed time he went to the library, and it was there he came upon Fermat’s Last Theorem. He became so inspired by the problem that he forgot about suicide and his death wish subsided. As an act of gratitude to the problem that saved his life he left in his will the equivalent of $2 million to whoever could solve the problem - the Wolfskehl Prize.
This reward generated renewed interest in the problem at the start of this century, but none of these new attempts solved the problem. However, during the 300 years of failed attempts new and interesting mathematics has been generated, which has spawned new areas of research. The Last Theorem touches on many  areas of mathematics, and the history of Fermat’s Last Theorem, which I suppose has its origins with Pythagoras, is, in a way, the history of mathematics.

But in the 20th century couldn’t we use computers to prove the Last Theorem?

The Last Theorem says that of all the infinity of numbers none will fit Fermat’s equation. A computer could check the first thousand numbers, and none might fit, but what about the next number? A computer could check the first million numbers, but what about a million and one. No matter how high a computer checks it will never get to infinity, and so it cannot prove the Last Theorem. Brute force checking is not adequate - what you need is a logical proof, a reason.

But how could you ever prove that something is true for infinity?

One method is a type of mathematical domino toppling. If you want to knock down an infinite row of dominos you have to arrange them carefully, such that if you topple the first one, then you will topple all of them. Mathematically you have to prove that a theorem is true for the first number, and then show that if it is true for any number, then it must be true for the next one.

Mathematicians are thought to be strange people. Could you explain what they are like?

Very few people know what mathematicians do, let alone what motivates them. Few people realise that mathematics is a subject that requires determination, imagination and creativity, and that mathematicians wander into in abstract territories that have yet to be explored. They grapple with infinity, they discover new theorems, and they might even stumble upon a completely new type of number.

In the story that I tell, the central mathematician who tries to prove Fermat’s Last Theorem is Andrew Wiles. He is an extraordinarily brilliant mathematician, therefore he is determined, imaginative, creative and courageous. He has a passion for his subject and a particular obsession for Fermat's Last Theorem.

At the same time he has broader range of interests - he has a family, he was a keen athlete, he is well read and eloquent. And this is typical for a mathematician. Mathematicians are not nerdy, they are just misunderstood by the rest of the world.

Andrew Wiles is the hero of the book - how did he come across the Last Theorem?

It was 1963, Wiles was just ten years old, and he was in his local public library when he found a book about Fermat’s Last Theorem. The problem seemed relatively simple, and even though nobody had been able to prove it for three centuries, Wiles swore that he would prove it. His teachers told him to get on with his homework. Then when he went to university his professors told him to concentrate on his coursework. Then when he became a professor himself his colleagues told him to work on mainstream problems, because otherwise he could jeopardize his whole career. But Wiles held on to his childhood dream. He refused to give up.

When did Wiles make his crucial breakthrough?

Wiles is English, but in the 1980s he took up a professorship at Princeton. Then in 1986 he heard that the Last Theorem had been linked to something called the Taniyama-Shimura conjecture - another mathematical problem. The upshot was that if you could prove the Taniyama-Shimura conjecture, then you would immediately prove Fermat’s Last Theorem. However, the Taniyama-Shimura conjecture was itself an old and difficult problem, and so many people thought that this alternative route to a proof was equally impossible.

What did Wiles do then?

He decided to prove the Taniyama-Shimura conjecture and hence prove Fermat’s Last Theorem. The first decision he made was to work in complete secrecy. He was worried that if somebody knew what he was up to, then they might steal his idea and beat him to the prize. He isolated himself for seven years, and only his wife knew what he was doing. He told her on their honeymoon. Then in order not to arouse suspicion he began to publish some previous research which had been gathering dust. This false trail meant that his colleagues assumed that he was working on conventional problems. Wiles’ ordeal required immense stamina and courage, but also an element cunning and deceit.

But what did he do for seven years?

Wiles was building a logical argument. Step by step he was constructing one of the most elaborate proofs in the history of mathematics. Curiously he was pulling together many of the ideas that failed attempts had generated, in order to solve the problem. Sometimes he was inventing new techniques, and at other times he was looking back at old mathematics. One of Wiles’ inspirations came from Evariste Galois, a nineteenth century mathematician.

Galois was a prodigy. As a teenager he invented many mathematical techniques, but his work was ignored by the establishment, because Galois was a republican, at a time when the state was moving towards the King. After being rejected, Galois gave up on mathematics, and eventually fell in love with Stephanie du Motel. Unfortunately Stephanie was already engaged to the finest shot in all of France, and he challenged Galois to a duel at dawn. Galois realized that he would probably die, and stayed up all night writing down all his theorems. The next day he did indeed die, and his brother took Galois’ notes to the Academy. Eventually his ideas were appreciated, and ultimately they became fundamental to Wiles’ proof.

When Wiles started his seven years of secrecy did he have a clear strategy?

No, not at all. Wiles describes his experience of doing mathematics as a journey through a dark unexplored mansion. You enter the first room and it’s completely dark. You stumble around, bumping into the furniture. After six months or so, you find the light switch, and suddenly everything is illuminated. Then you move into the next room and spend another six months in the dark. Although each of these breakthroughs can be momentary, they are the culmination of many months of stumbling around in the dark.

When Wiles finally found the final light in 1993, what was the reaction?

Wiles returned to Cambridge, England to speak at a conference, but nobody realized that he was going to reveal a proof of the world’s most notorious mathematical problem. The mathematical community was astonished - this was the proof of the century. Wiles was on the front page of the New York Times, he was being interviewed on CNN, he was the world’s most famous mathematician. He had proved the Taniyama-Shimura conjecture, and hence the Last Theorem. He had apparently achieved his childhood dream.

But that wasn’t the end of the story.

All proofs have to be checked before they can be accepted. That summer a team of referees examined the proof line by line. Slowly an error began to emerge - a fundamental mistake which destroyed the whole proof. There is a saying in mathematics: “A problem worthy of attack, proves its worth by fighting back.” Fermat seemed to be having the last laugh. Wiles immediately locked himself away, and attempted to fix the error.

How did the community react to the error?

There was some criticism of Wiles, because he refused to release the manuscript, and the error. You can understand his situation - he had worked for seven years, and there was the danger that somebody would look at his proof, fix it, and steal the glory. After a year of trying and failing to fix the error, the pressure to release the manuscript was enormous, but just when he was about to give up, Wiles fixed his proof.

I made a TV documentary about this with a colleague, John Lynch. It was shown on the BBC, and also on PBS as part of the NOVA series, and when Wiles recounted the final moments of his proof he actually broke down and was overcome with emotion. This was the culmination of a lifelong ambition, and the end of a year of torment.

When Wiles’s proof was disproved, he went from ecstasy to agony. The part of the book that recounts when he came to terms with this disaster is particularly emotional, isn’t it?

The book is about a human struggle, a quest. Like any other great quest there are moments when the hero is confronted by disaster, and this is what happened to Wiles in 1993, just weeks after he thought that he had achieved his goal. I think that we can all sympathise with this very human experience, the experience of utter devastation. However, I think that people will be surprised that this is possible for a mathematician, because most people have a misconception that the subject is dry and boring.

Lots of mathematicians believe that Fermat was playing games… What’s your opinion? Was it possible for a man in 1637, taking into account the mathematical methods available then, to find a solution to the problem?

Wiles’s proof could not possibly be the same as Fermat’s. Wiles’s proof relies on some of the most sophisticated ideas of the 20th century, and so there are two possibilities. Either Fermat did not have a proof, and had simply made a mistake. Or Fermat had an elegant 17th century proof of the theorem. There are mathematicians who believe that the latter is possible, and that the proof is out there somewhere, and they continue to search for the original proof of Fermat’s Last Theorem.

But I do not think that Fermat was playing games. I do not think that his marginal note was a joke. I do not think that he ever expected anybody to read his marginal jottings after his death.

What is Wiles doing now?

Bearing in mind that he worked in secrecy for seven years, it is difficult to know what he is up to at the moment. But he does have a dilemma. Now that he has solved the greatest problem in mathematics, he will never find another problem which will mean the same to him. His mind is now free, he has achieved his dream, but there is also a certain emptiness.

Isaac Newton said: ‘If I have seen further it is by standing on the shoulders of giants’. That is, future development relies on the foundation of past inventions and discoveries. Does Wiles's work tell us that Newton was right?

Absolutely. Wiles's proof is over 100 pages long and it draws upon the work of many other mathematicians through the century. One mathematician told me that if you were lost on a desert island with this one proof, then you would have a good overview of all number theory. Wiles's genius is that he was able to draw together the diverse ideas of the mathematicians around him and before him to construct a proof of Fermat's Last Theorem.

What will Wiles's accomplishment bring to the future of mathematics?

Wiles's proof of Fermat's Last Theorem is in itself a bit of a dead end. However, the proof includes a proof of the Taniyama- Shimura Conjecture, and it is this part of his work that will have major repercussions. Within pure mathematics it opens up whole new areas for research and exploration. We will have to wait and see whether Wiles's work has any applications in the real world.

Perhaps equally important is the fact that Wiles's work has inspired a whole generation of mathematicians and will have encouraged a new generation of youngsters to enter the subject. He has brought mathematics to the attention of the world.

In your book it is said that mathematicians are feeling somewhat empty because Fermat's Last Theorem has now been solved. Is it really true that mathematicians are missing the Last Theorem?

Fermat's Last Theorem was the greatest problem in mathematics, and so mathematicians naturally wanted to solve it. However, once is has been solved you need something else to replace it. Some other problem to act as a focus for mathematics. There is no shortage of difficult mathematical problems, perhaps the most important one being the Riemann Hypothesis. However, most of these problems are not easy to explain to the public and neither do they have the history or romance of Fermat's Last Theorem.

Interestingly, a committee of mathematicians, including Andrew Wiles, offered 7 prizes of 1 million dollars each for the solution of 7 mathematical problems. You can find out more about the problems at the Clay Foundation website. Good luck.