Fermat's Last Theorem (16 page)

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.

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.

Of all the European countries France displayed the most chauvinistic attitude towards educated women, declaring that mathematics was unsuitable for women and beyond their mental capacity. Although the salons of Paris dominated the mathematical world for most of the eighteenth and nineteenth centuries, only one woman managed to escape the constraints of French society and establish herself as a great number theorist. Sophie Germain revolutionised the study of Fermat's Last Theorem and made a contribution greater than any of the men who had gone before her.

Sophie Germain was born on 1 April 1776, the daughter of a merchant, Ambroise-François Germain. Outside of her work, her
life was to be dominated by the turmoils of the French Revolution – the year she discovered her love of numbers the Bastille was stormed, and her study of calculus was shadowed by the Reign of Terror. Although her father was financially successful, Sophie's family were not members of the aristocracy.

Although ladies of Germain's social background were not actively encouraged to study mathematics, they were expected to have sufficient knowledge of the subject in order to be able to discuss the topic should it arise during polite conversation. To this end a series of textbooks were written to help young women get to grips with the latest developments in mathematics and science. Francesco Algarotti was the author of
Sir Isaac Newton's Philosophy Explain'd for the Use of Ladies.
Because Algarotti believed that women were only interested in romance, he attempted to explain Newton's discoveries through the flirtatious dialogue between a Marquise and her interlocutor. For example, the interlocutor outlines the inverse square law of gravitational attraction, whereupon the Marquise gives her own interpretation on this fundamental law of physics: ‘I cannot help thinking … that this proportion in the squares of the distances of places … is observed even in love. Thus after eight days' absence love becomes sixty-four times less than it was the first day.'