Fermat's Last Theorem (11 page)

In the margin of his
Arithmetica
, next to Problem 8, he made a note of his observation:

Cubem autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere.

It is impossible for a cube to be written as a sum of two cubes or a fourth power to be written as the sum of two fourth powers or, in general, for any number which is a power greater than the second to be written as a sum of two like powers.

Among all the possible numbers there seemed to be no reason why at least one set of solutions could not be found, yet Fermat stated that nowhere in the infinite universe of numbers was there a ‘Fermatean triple'. It was an extraordinary claim, but one which Fermat believed he could prove. After the first marginal note outlining the theory, the mischievous genius jotted down an additional comment which would haunt generations of mathematicians:

Cuius rei demonstrationem mirabilem sane detexi hanc marginis exiguitas non caperet.

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

This was Fermat at his most infuriating. His own words suggest that he was particularly pleased with this ‘truly marvellous' proof, but he had no intention of bothering to write out the detail of the argument, never mind publishing it. He never told anyone about his proof, and yet despite his combination of indolence and modesty Fermat's Last Theorem, as it would later be called, would become famous around the world for centuries to come.

Fermat's notorious discovery happened early in his mathematical career, in around 1637. Some thirty years later, while carrying out his judicial duties in the town of Castres, Fermat was taken seriously ill. On 9 January 1665, he signed his last
arrêt
, and three days later he died. Still isolated from the Parisian school of mathematics and not necessarily fondly remembered by his frustrated correspondents, Fermat's discoveries were at risk of being lost forever. Fortunately Fermat's eldest son, Clément-Samuel, who appreciated the significance of his father's hobby, was determined that his discoveries should not be lost to the world. It is thanks to his efforts that we know anything at all about Fermat's remarkable breakthroughs in number theory and, in particular, if it were not for Clément-Samuel, the enigma known as Fermat's Last Theorem would have died with its creator.

Clément-Samuel spent five years collecting his father's notes and letters, and examining the jottings in the margins of his copy of the
Arithmetica.
The marginal note referring to Fermat's Last Theorem was just one of many inspirational thoughts scribbled in the book, and Clément-Samuel undertook to publish these annotations in a special edition of the
Arithmetica.
In 1670 at Toulouse he brought out
Diophantus' Arithmetica Containing Observations by P. de Fermat
. Alongside Bachet's original Greek and Latin translations were forty-eight observations made by Fermat, one of which was to become known as Fermat's Last Theorem.

Once Fermat's
Observations
reached the wider community, it was clear that the letters he had sent to colleagues represented mere morsels from a treasure trove of discovery. His personal notes contained a whole series of theorems. Unfortunately these were accompanied either with no explanation at all or with only a slight hint of the underlying proof. There were just enough tantalising glimpses of logic to leave mathematicians in no doubt that Fermat had proofs, but filling in the details was left as a challenge for them to take up.

Leonhard Euler, one of the greatest mathematicians of the eighteenth century, attempted to prove one of Fermat's most elegant observations, a theorem concerning prime numbers. A prime number is one which has no divisors – no number will divide into it without a remainder, except for 1 and the number itself. For instance, 13 is a prime number, but 14 is not. Nothing will divide into 13, but 2 and 7 will divide into 14. All prime numbers (except 2) can be put into two categories; those which equal 4
n
+ 1 and those which equal 4
n
– 1, where
n
equals some number. So 13 is in the former group (4 × 3 + 1), whereas 19 is in the latter group (4 × 5 – 1). Fermat's prime theorem claimed that the first type of primes were always the sum of two squares (13 = 2
²
+ 3
²
), whereas the second type could never be written in this way (19 = ?
²
+ ?
²
). This property of primes is beautifully simple, but trying to prove that it is true for every single prime number turns out to be remarkably difficult. For Fermat it was just one of many private proofs. The challenge for Euler was to rediscover Fermat's proof. Eventually in 1749, after seven years work and almost a century after Fermat's death, Euler succeeded in proving this prime number theorem.

Fermat's panoply of theorems ranged from the fundamental to the simply amusing. Mathematicians rank the importance of theorems according to their impact on the rest of mathematics. First, a theorem is considered important if it has a universal truth, that is to say, if it applies to an entire group of numbers. In the case of the prime number theorem, it is true not for just some prime numbers,
but for all prime numbers. Second, theorems should reveal some deeper underlying truth about the relationship between numbers. A theorem can be the springboard for generating a whole host of other theorems, even inspiring the development of whole new branches of mathematics. Finally, a theorem is important if entire areas of research can be hindered for the lack of one logical link. Many mathematicians have cried themselves to sleep knowing that they could achieve a major result if only they could establish one missing link in their chain of logic.

Because mathematicians employ theorems as stepping stones to other results, it was essential that every single one of Fermat's theorems be proved. Just because Fermat said he had a proof of a theorem it could not be accepted at face value. Before it could be used, each theorem had to be proved with ruthless rigour, otherwise the consequences could have been disastrous. For example, imagine that mathematicians had accepted one of Fermat's theorems. It would then be incorporated as a single element in a whole series of other larger proofs. In due course these larger proofs would be incorporated into even larger proofs, and so on. Ultimately hundreds of theorems could come to rely on the truth of the original unchecked theorem. However, what if Fermat had made a mistake and the unchecked theorem was in fact flawed? All these other theorems which incorporated it would also be flawed, and vast areas of mathematics would collapse. Theorems are the foundations of mathematics, because once their truth has been established other theorems can safely be built on top of them. Unsubstantiated ideas are infinitely less valuable and are referred to as conjectures. Any logic which relies on a conjecture is itself a conjecture.

Fermat said he had a proof for every one of his observations, so for him they were theorems. However, until the community at large could rediscover the individual proofs each one could only be
considered a conjecture. 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. Three centuries of effort failed to find a proof, and this led to its notoriety as the most demanding riddle in mathematics. However, 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. 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.'

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.'