Fermat's Last Theorem

by Simon Singh

  Intuitively this enormously simplifies the problem, because you can ignore those equations which involve a value of n that is not a prime number. The number of equations remaining is now vastly reduced. For example, for the values of n up to 20, there are only six values which need to be proved:

  If one can prove Fermat’s Last Theorem for just the prime values of n, then the theorem is proved for all values of n. If one considers all whole numbers, then it is obvious that there are infinitely many. If one considers just the prime numbers, which are only a small fraction of all the whole numbers, then surely the problem is much simpler?

  Intuition would suggest that if you begin with an infinite quantity and then remove the bulk of it, then you would expect to be left with something finite. Unfortunately intuition is not the arbiter of truth in mathematics, but rather logic. In fact, it is possible to prove that the list of primes is never-ending. Therefore, despite being able to ignore the vast majority of equations relating to non-prime values of n, the remaining equations relating to prime values of n are still infinite in number.

  The proof that there is an infinity of primes dates all the way back to Euclid, and is one of the classic arguments of mathematics. Initially Euclid assumes that there is a finite list of known prime numbers, and then shows that there must exist an infinite number of additions to this list. There are N prime numbers in Euclid’s finite list, which are labelled P1, P2, P3, …,PN. Euclid can then generate a new number QA such that

  This new number QA is either prime or not prime. If it is prime then we have succeeded in generating a new, bigger prime number, and therefore our original list of primes was not complete. On the other hand, if QA is not prime, then it must be perfectly divisible by a prime. This prime cannot be one of the known primes because dividing QA by any of the known primes will inevitably lead to a remainder of 1. Therefore there must be some new prime, which we can call PN + 1.

  We have now arrived at the stage where either QA is a new prime or we have another new prime PN+1. Either way we have added to our original list of primes. We can now repeat the process, including our new prime (PN+1 or QA) in our list, and generate some new number QB. Either this new number will be yet another new prime, or there will have to be some other new prime PN+2 that is not on our list of known primes. The upshot of the argument is that, however long our list of prime numbers, it is always possible to find a new one. Therefore the list of primes is never-ending and infinite.

  But how can something which is undeniably smaller than an infinite quantity also be infinite? The German mathematician David Hilbert once said: ‘The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite.’ To resolve the paradox of the infinite it is necessary to define what is meant by infinity. Georg Cantor, who worked alongside Hilbert, defined infinity as the size of the never-ending list of counting numbers (1, 2, 3, 4, …). Consequently anything which is comparable in size is equally infinite.

  By this definition the number of even counting numbers, which would intuitively appear to be smaller, is also infinite. It is easy to demonstrate that the quantity of counting numbers and the quantity of even numbers are comparable because we can pair off each counting number with a corresponding even number:

  If every member of the counting numbers list can be matched up with a member of the even numbers list then the two lists must be the same size. This method of comparison leads to some surprising conclusions, including the fact that there are an infinite number of primes. Although Cantor was the first person to tackle infinity in a formal way, he was initially heavily criticised by the mathematical community for his radical definition. Towards the end of his career the attacks became increasingly personal and this resulted in Cantor suffering mental illness and severe depression. Eventually, after his death, his ideas became widely accepted as the only consistent, accurate and powerful definition of infinity. As a tribute Hilbert said: ‘No one shall drive us from the paradise Cantor has created for us.’

  Hilbert went on to create an example of infinity, known as Hilbert’s Hotel, which clearly illustrates its strange qualities. This hypothetical hotel has the desirable attribute of having an infinite number of rooms. One day a new guest arrives and is disappointed to learn that, despite the hotel’s infinite size, all the rooms are occupied. Hilbert, the clerk, thinks for a while and then reassures the new arrival that he will find an empty room. He asks all his current guests to move to the next room, so that the guest in room 1 moves to room 2, the guest in room 2 moves to room 3, and so on. Everybody who was in the hotel still has a room, which allows the new arrival to slip into the vacant room 1. This shows that infinity plus one equals infinity.

  The following night Hilbert has to deal with a much greater problem. The hotel is still full when an infinitely large coach arrives with an infinite number of new guests. Hilbert remains unperturbed and rubs his hands at the thought of infinitely more hotel bills. He asks all his current guests to move to the room which is double the number of their current room. So the guest in room 1 moves to room 2, the guest in room 2 moves to room 4, and so on. Everybody who was in the hotel still has a room and yet an infinite number of rooms, all the odd ones, have been vacated for the new arrivals. This shows that double infinity is still infinity.

  Hilbert’s Hotel seems to suggest that all infinities are as large as each other, because various infinities seem to be able to squeeze into the same infinite hotel – the infinity of even numbers can be matched up and compared with the infinity of all counting numbers. However, some infinities are indeed bigger than others. For example, any attempt to pair every rational number with every irrational number ends in failure, and in fact it can be proved that the infinite set of irrational numbers is larger than the infinite set of rational numbers. Mathematicians have had to develop a whole system of nomenclature to deal with the varying scales of infinity and conjuring with these concepts is one of today’s hottest topics.

  Although the infinity of primes dashed hopes for an early proof of Fermat’s Last Theorem, a countless supply of prime numbers does have more positive implications in other areas such as espionage and the evolution of insects. Before returning to the quest for a proof of Fermat’s Last Theorem it is worth briefly investigating the uses and abuses of primes.

  Prime number theory is one of the few areas of pure mathematics that has found a direct application in the real world, namely in cryptography. Cryptography involves scrambling secret messages so that they can only be unscrambled by the receiver and not by anybody else who might intercept them. The scrambling process requires the use of a secret key, and traditionally unscrambling the message simply requires the receiver to apply the key in reverse. With this procedure the key is the weakest link in the chain of security. First, the receiver and the sender must agree on the details of the key and the exchange of this information is a risky process. If the enemy can intercept the key being exchanged, then they can unscramble all subsequent messages. Second, the keys must be regularly changed in order to maintain security, and each time this happens there is a risk of the new key being intercepted.

  The problem of the key revolves around the fact that applying it one way will scramble the message, and applying it in reverse unscrambles the message – unscrambling a message is almost as easy as scrambling it. However, experience tells us that there are many everyday situations when unscrambling is far harder than scrambling – it is relatively easy to scramble an egg, but to unscramble it is far harder.

  In the 1970s Whitfield Diffie and Martin Hellman came up with the idea of looking for a mathematical process which was easy to perform in one direction but incredibly difficult to perform in the opposite direction. Such a process would provide a perfect key. For example, I could have my own two-part key, and publish the scrambling half of it in a public directory. Then anybody could send me scrambled
messages, but only I would know the unscrambling half of the key. Although everyone would have knowledge of the scrambling part of the key, it bears no relation to the unscrambling part of the key.

  In 1977 Ronald Rivest, Adi Shamir and Leonard Adleman, a team of mathematicians and computer scientists at the Massachusetts Institute of Technology, realised that prime numbers were the ideal basis for an easy-scramble/hard-unscramble process. In order to make my own personal key I would take two huge prime numbers, each one containing up to 80 digits, and then multiply them together to achieve an even larger non-prime number. In order to scramble messages all that is required is knowledge of the large non-prime number, whereas to unscramble the message you would need to know the two original prime numbers which were multiplied together, known as the prime factors. I can now publish the large non-prime number, the scrambling half of the key, and keep the two prime factors, the unscrambling half of the key, to myself. Importantly, even though everybody knows the large non-prime number, they would have immense difficulty in working out the two prime factors.

  Taking a simpler example, I could hand out the non-prime number 589, which would enable everyone to scramble messages to me. I would keep the two prime factors of 589 secret, so that only I could unscramble the messages. If others could work out the two prime factors then they too could unscramble my messages, but even with this small number it is not obvious what the two prime factors are. In this case it would only take a few minutes on a desktop computer to figure out that the prime factors are actually 31 and 19 (31 × 19 = 589), and so my key would not remain secure for very long.

  However, in reality the non-prime number which I would publish would have over a hundred digits, which makes the task of finding its prime factors effectively impossible. Even if the world’s most powerful computers were used to split this huge non-prime number (the scrambling key) into its two prime factors (the unscrambling key) it would take several years to achieve the answer. Therefore, to foil foreign spies, I merely have to change my key on an annual basis. Once a year I announce my new giant non-prime number, and anybody who wants to try and unscramble my messages would then have to start all over again trying to compute the two prime factors.

  As well as finding a role in espionage, prime numbers also appear in the natural world. The periodical cicadas, most notably Magicicada septendecim, have the longest life-cycle of any insect. Their unique life-cycle begins underground, where the nymphs patiently suck the juice from the roots of trees. Then, after 17 years of waiting the adult cicadas emerge from the ground, swarm in vast numbers and temporarily swamp the landscape. Within a few weeks they mate, lay their eggs and die.

  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.

  One theory suggests that the cicada has a parasite which also goes through a lengthy life-cycle and which the cicada is trying to avoid. If the parasite has a life-cycle of, say, 2 years then the cicada wants to avoid a life-cycle which is divisible by 2, otherwise the parasite and the cicada will regularly coincide. Similarly, if the parasite has a life-cycle of 3 years then the cicada wants to avoid a life-cycle which is divisible by 3, otherwise the parasite and the cicada will once again regularly coincide. Ultimately, to avoid meeting its parasite the cicadas’ best strategy is to have a long life-cycle lasting a prime number of years. Because nothing will divide into 17, Magicicada septendecim will rarely meet its parasite. If the parasite has a 2-year life-cycle they will only meet every 34 years, and if it has a longer life-cycle, say 16 years, then they will only meet every 272 (16 × 17) years.

  In order to fight back, the parasite only has two life-cycles which will increase the frequency of coincidences – the annual cycle and the same 17-year cycle as the cicada. However, the parasite is unlikely to survive reappearing 17 years in a row, because for the first 16 appearances there will be no cicadas for it to parasitise. On the other hand, in order to reach the 17-year life-cycle, the generations of parasites would first have to evolve through the 16-year life-cycle. This would mean at some stage of evolution the parasite and cicada would not coincide for 272 years! In either case the cicadas long prime life-cycle protects it.

  This might explain why the alleged parasite has never been found! In the race to keep up with the cicada, the parasite probably kept extending its life-cycle until it hit the 16-year hurdle. Then it failed to coincide for 272 years, by which time the lack of coinciding with cicadas had driven it to extinction. The result is a cicada with a 17-year life cycle, which it no longer needs because its parasite no longer exists.

  Monsieur Le Blanc

  By the beginning of the nineteenth century, Fermat’s Last Theorem had already established itself as the most notorious problem in number theory. Since Euler’s breakthrough there had been no further progress, but a dramatic announcement by a young Frenchwoman was to reinvigorate the pursuit of Fermat’s lost proof. Sophie Germain lived in an era of chauvinism and prejudice, and in order to conduct her research she was forced to assume a false identity, study in terrible conditions and work in intellectual isolation.

  Over the centuries women have been discouraged from studying mathematics, but despite the discrimination there have been several female mathematicians who fought against the establishment and indelibly forged their names in the annals of mathematics. 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. Pythagoras is known as the ‘feminist philosopher’ because he actively encouraged women scholars, Theano being just one of the twenty-eight sisters in the Pythagorean Brotherhood.

  In later centuries the likes of Socrates and Plato would continue to invite women into their schools, but it was not until the fourth century AD that a woman mathematician founded her own influential school. 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. Mathematicians who had been stuck for months on a particular problem would write to her seeking a solution, and Hypatia rarely disappointed her admirers. She was obsessed by mathematics and the process of logical proof, and when asked why she never married she replied that she was wedded to the truth. Ultimately her devotion to the cause of rationalism caused her downfall, when Cyril, the patriarch of Alexandria, began to oppress philosophers, scientists and mathematicians, whom he called heretics. The historian Edward Gibbon provided a vivid account of what happened after Cyril had plotted against Hypatia and turned the masses against her:

  On a fatal day, in the holy season of Lent, Hypatia was torn from her chariot, stripped naked, dragged to the church, and inhumanely butchered by the hands of Peter the Reader and a troop of savage and merciless fanatics; her flesh was scraped from her bones with sharp oyster-shells, and her quivering limbs were delivered to the flames.

  Soon after the death of Hypatia mathematics entered a period of stagnation and it was not until after the Renaissance that another woman made her name as a mathematician. Maria Agnesi was born in Milan in 1718 and, like Hypatia, was the daughter of a mathematician. She was acknowledged to be one of the finest mathematicians in Europe, particularly famous for her treatises on the tangents to curves. In Italian, curves were called versiera, a word derived from the Latin vertere, ‘to turn’, but it was also an abbreviation for avversiera, or ‘wife of the Devil’. A curve studied by Agnesi (versiera Agnesi) was mistranslated into English as the ‘witch of Agnesi’, and in time the mathematician herself was referred to by the same title.

  Although mathematicians across Europe acknowledged Agnesi’s ability, many academic institutions, in particular the French Academy, refused to give her a
research post. Institutionalised discrimination against women continued right through to the twentieth century, when Emmy Noether, described by Einstein as ‘the most significant creative mathematical genius thus far produced since the higher education of women began’, was denied a lectureship at the University of Göttingen. The majority of the faculty argued: ‘How can it be allowed that a woman become a Privatdozent? Having become a Privatdozent, she can then become a professor and a member of the University Senate …. What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?’ Her friend and mentor David Hilbert replied: ‘Meine Herren, I do not see that the sex of the candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bathhouse.’

  Later her colleague Edmund Landau was asked whether Noether was indeed a great woman mathematician, to which he replied: ‘I can testify that she is a great mathematician, but that she is a woman, I cannot swear.’

  In addition to suffering discrimination Noether had much else in common with other women mathematicians through the centuries, such as the fact that she too was the daughter of a mathematics professor. Many mathematicians, of both genders, are from mathematical families, giving rise to light-hearted rumours of a mathematical gene, but in the case of women the percentage is particularly high. The probable explanation is that most women with potential were never exposed to the subject or encouraged to pursue it, whereas those born to professors could hardly avoid being immersed in the numbers. Furthermore, Noether, like Hypatia, Agnesi and most other women mathematicians, never married, largely because it was not socially acceptable for women to pursue such careers and there were few men who were prepared to wed brides with such controversial backgrounds. The great Russian mathematician Sonya Kovalevsky is an exception to this rule, inasmuch as she arranged a marriage of convenience to Vladimir Kovalevsky, a man who was agreeable to a platonic relationship. For both parties the marriage allowed them to escape their families and concentrate on their researches, and in Sonya’s case travelling alone around Europe was much easier once she was a respectable married woman.