Fermat's Last Theorem

by Simon Singh

  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.’

  Not surprisingly this gallant genre of books was not responsible for inspiring Sophie Germain’s interest in mathematics. The event that changed her life occurred one day when she was browsing in her father’s library and chanced upon Jean-Etienne Montucla’s book History of Mathematics. The chapter that caught her imagination was Montucla’s essay on the life of Archimedes. His account of Archimedes’ discoveries was undoubtedly interesting, but what particularly kindled her fascination was the story surrounding his death. 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.

  Germain concluded that if somebody could be so consumed by a geometric problem that it could lead to their death, then mathematics must be the most captivating subject in the world. She immediately set about teaching herself the basics of number theory and calculus, and soon she was working late into the night, studying the works of Euler and Newton. This sudden interest in such an unfeminine subject worried her parents. A friend of the family, Count Guglielmo Libri-Carrucci dalla Sommaja, told how Sophie’s father confiscated her candles and clothes and removed any heating in order to discourage her from studying. Only a few years later in Britain, the young mathematician Mary Somerville would also have her candles confiscated by her father who maintained that ‘we must put a stop to this, or we shall have Mary in a strait-jacket one of these days’.

  In Germain’s case she responded by maintaining a secret cache of candles and wrapping herself in bed-clothes. Libri-Carrucci wrote that the winter nights were so cold that the ink froze in the inkwell but Sophie continued regardless. She was described by some people as shy and awkward, but she was also immensely determined and eventually her parents relented and gave Sophie their blessing. Germain never married and throughout her career her father funded her research. For many years Germain continued to study alone because there were no mathematicians in the family who could introduce her to the latest ideas and her tutors refused to take her seriously.

  Then, in 1794, the Ecole Polytechnique opened in Paris. It was founded as an academy of excellence to train mathematicians and scientists for the nation. This would have been an ideal place for Germain to develop her mathematical skills except for the fact that it was an institution reserved only for men. Her natural shyness prevented her from confronting the academy’s governing body, so instead she resorted to covertly studying at the Ecole by assuming the identity of a former student at the academy, Monsieur Antoine-August Le Blanc. The academy’s administration was unaware that the real Monsieur Le Blanc had left Paris and continued to print lecture notes and problems for him. Germain managed to obtain what was intended for Le Blanc and each week she would submit answers to the problems under her new pseudonym. Everything was going to plan until a couple of months later when the supervisor of the course, Joseph-Louis Lagrange, could no longer ignore the brilliance of Monsieur Le Blanc’s answer sheets. Not only were Monsieur Le Blanc’s solutions marvellously ingenious, but they showed a remarkable transformation in a student who had previously been notorious for his abysmal calculations. Lagrange, who was one of the finest mathematicians of the nineteenth century, requested a meeting with the reformed student and Germain was forced to reveal her true identity. Lagrange was astonished and pleased to meet the young woman and became her mentor and friend. At last Sophie Germain had a teacher who could inspire her, and with whom she could be open about her skills and ambitions.

  Germain grew in confidence and she moved from solving problems in her coursework to studying unexplored areas of mathematics. Most importantly she became interested in number theory and inevitably she came to hear of Fermat’s Last Theorem. She worked on the problem for several years, eventually reaching the stage where she believed she had made an important breakthrough. She needed to discuss her ideas with a fellow number theorist and decided that she would go straight to the top and consult the greatest number theorist in the world, the German mathematician Carl Friedrich Gauss.

  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’. Germain had first encountered his work through studying his masterpiece Disquisitiones arithmeticae, the most important and wide-ranging treatise since Euclid’s Elements. Gauss’s work influenced every area of mathematics, but strangely enough he never published anything on Fermat’s Last Theorem. In one letter he even displayed contempt for the problem. His friend the German astronomer Heinrich Olbers had written to Gauss encouraging him to compete for a prize which had been offered by the Paris Academy for a solution to Fermat’s challenge: ‘It seems to me, dear Gauss, that you should get busy about this.’ Two weeks later Gauss replied, ‘I am very much obliged for your news concerning the Paris prize. But I confess that Fermat’s Last Theorem as an isolated proposition has very little interest for me, for I could easily lay down a multitude of such propositions, which one could neither prove nor disprove.’ Gauss was entitled to his opinion, but Fermat had clearly stated that a proof existed and even the subsequent failed attempts to find the proof had generated innovative new techniques, such as proof by ‘infinite descent’ and the use of imaginary numbers. Perhaps in the past Gauss had tried and failed to make any impact on the problem, and his response to Olbers was merely a case of intellectual sour grapes. Nonetheless, when he received Germain’s letters he was sufficiently impressed by her breakthrough that he temporarily forgot his ambivalence towards Fermat’s Last Theorem.

  Seventy-five years earlier Euler had published his proof for the case n = 3, and ever since mathematicians had been trying in vain to prove other individual cases. However, Germain adopted a new strategy and described to Gauss a so-called general approach to the problem. In other words, her immediate goal was not to prove one partic
ular case, but to say something about many cases at once. In her letter to Gauss she outlined a calculation which focused on a particular type of prime number p such that (2p + 1) is also prime. Germain’s list of primes includes 5, because 11 (2 × 5 + 1) is also prime; but it does not include 13, because 27 (2 × 13 + 1) is not prime.

  For values of n equal to these Germain primes, she used an elegant argument to show that there were probably no solutions to the equation xn + yn = zn. By ‘probably’ Germain meant that it was unlikely that any solutions existed, because if there was a solution then either x, y or z would be a multiple of n, and this would put a very tight restriction on any solutions. Her colleagues examined her list of primes one by one trying to prove that x, y or z could not be a multiple of n, thereby showing that for that particular value of n there could be no solutions.

  In 1825 her method claimed its first complete success thanks to Gustav Lejeune-Dirichlet and Adrien-Marie Legendre, two mathematicians a generation apart. Legendre was a man in his seventies who had lived through the political turmoil of the French Revolution. His failure to support the government candidate for the Institut National led to the stopping of his pension, and by the time he made his contribution to Fermat’s Last Theorem he was destitute. On the other hand, Dirichlet was an ambitious young number theorist who had only just turned twenty. Both of them independently were able to prove that the case n = 5 has no solutions, but they based their proofs on, and owed their success to, Sophie Germain.

  Fourteen years later the French made another breakthrough. Gabriel Lamé made some further ingenious additions to Germain’s method and proved the case for the prime n = 7. Germain had shown numbers theorists how to destroy an entire section of prime cases and now it was up to the combined efforts of her colleagues to continue proving Fermat’s Last Theorem one case at a time.

  Germain’s work on Fermat’s Last Theorem was to be her greatest contribution to mathematics but initially she was not credited for her breakthrough. When Germain wrote to Gauss she was still in her twenties, and although she had gained a reputation in Paris she feared that the great man would not take her seriously because of her gender. In order to protect herself Germain resorted once again to her pseudonym, signing her letters as Monsieur Le Blanc.

  Her fear and respect for Gauss is shown in one of her letters to him: ‘Unfortunately, the depth of my intellect does not equal the voracity of my appetite, and I feel a kind of temerity in troubling a man of genius when I have no other claim to his attention than an admiration necessarily shared by all his readers.’ Gauss, unaware of his correspondent’s true identity, attempted to put Germain at ease and replied: ‘I am delighted that arithmetic has found in you so able a friend.’

  Germain’s contribution may have been forever wrongly attributed to the mysterious Monsieur Le Blanc were it not for the Emperor Napoleon. In 1806 Napoleon was invading Prussia and the French army was storming through one German city after another. Germain feared that the fate that befell Archimedes might also take the life of her other great hero Gauss, so she sent a message to her friend General Joseph-Marie Pernety, who was in charge of the advancing forces. She asked him to guarantee Gauss’s safety, and as a result the general took special care of the German mathematician, explaining to him that he owed his life to Mademoiselle Germain. Gauss was grateful but surprised, for he had never heard of Sophie Germain.

  The game was up. In Germain’s next letter to Gauss she reluctantly revealed her true identity. Far from being angry at the deception, Gauss wrote back to her with delight:

  But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal themselves only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarise herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and superior genius. Indeed nothing could prove to me in so flattering and less equivocal manner that the attractions of this science, which has enriched my life with so many joys, are not chimerical, as the predilection with which you have honoured it.

  Sophie Germain’s correspondence with Carl Gauss inspired much of her work, but in 1808 the relationship ended abruptly. Gauss had been appointed professor of astronomy at the University of Göttingen, his interest shifted from number theory to more applied mathematics, and he no longer bothered to return Germain’s letters. Without her mentor her confidence began to wane, and within a year she abandoned pure mathematics.

  Although she made no further contributions to proving Fermat’s Last Theorem, she did embark on an eventful career as a physicist, a discipline in which she would again excel only to be confronted by the prejudices of the establishment. Her most important contribution to the subject was ‘Memoir on the vibrations of elastic plates’, a brilliantly insightful paper which laid the foundations for the modern theory of elasticity. As a result of this research and her work on Fermat’s Last Theorem she received a medal from the Institut de France, and became the first woman who was not a wife of a member to attend lectures at the Academy of Sciences. Then towards the end of her life she re-established her relationship with Carl Gauss, who convinced the University of Göttingen to award her an honorary degree. Tragically, before the university could bestow the honour upon her, Sophie Germain died of breast cancer.

  All things considered she was probably the most profoundly intellectual woman that France has ever produced. And yet, strange as it may seem, when the state official came to make out the death certificate of this eminent associate and co-worker of the most illustrious members of the French Academy of Science, he designated her as a rentière-annuitant (a single woman with no profession) – not as a mathématicienne. Nor is this all. When the Eiffel Tower was erected, in which the engineers were obliged to give special attention to the elasticity of the materials used, there were inscribed on this lofty structure the names of seventy-two savants. But one will not find in this list the name of that daughter of genius, whose researches contributed so much towards establishing the theory of the elasticity of metals – Sophie Germain. Was she excluded from this list for the same reason that Agnesi was ineligible for membership in the French Academy – because she was a woman? It would seem so. If such, indeed, was the case, more is the shame for those who were responsible for such ingratitude towards one who had deserved so well of science, and who by her achievements had won an enviable place in the hall of fame.

  H.J. Mozans, 1913

  The Sealed Envelopes

  After the breakthrough of Sophie Germain the French Academy of Sciences offered a series of prizes, including a gold medal and 3,000 Francs to the mathematician who could finally put to rest the mystery of Fermat’s Last Theorem. As well as the prestige of proving Fermat’s Last Theorem there was now an immensely valuable reward attached to the challenge. The salons of Paris were full of rumours as to who was adopting which strategy and how close they were to announcing a result. Then, on 1 March 1847, the Academy held its most dramatic meeting ever.

  The proceedings describe how Gabriel Lamé, who had proved the case n = 7 some years earlier, took the podium in front of the most eminent mathematicians of the age and proclaimed that he was on the verge of proving Fermat’s Last Theorem. He admitted that his proof was still incomplete, but he outlined his method and predicted with relish that he would in the coming weeks publish a complete proof in the Academy’s journal.

  The entire audience was stunned, but as soon as Lamé left the floor Augustin Louis Cauchy, ano
ther of Paris’s finest mathematicians, asked for permission to speak. Cauchy announced to the Academy that he had been working along similar lines to Lamé, and that he too was about to publish a complete proof.

  Both Cauchy and Lamé realised that time was of the essence. Whoever would be first to submit a complete proof would receive the most prestigious and valuable prize in mathematics. Although neither of them had a complete proof, the two rivals were keen to somehow stake a claim and so just three weeks after they had made their announcements they deposited sealed envelopes at the Academy. This was a common practice at the time which enabled mathematicians to go on record without revealing the exact details of their work. If a dispute should later arise regarding the originality of ideas, then a sealed envelope would provide the evidence needed to establish priority.

  The anticipation built up throughout April as Cauchy and Lamé published tantalising but vague details of their proof in the proceedings of the Academy. Although the entire mathematical community was desperate to see the proof completed, many of them secretly hoped that it would be Lamé and not Cauchy who would win the race. By all accounts Cauchy was a self-righteous creature, a religious bigot and extremely unpopular with his colleagues. He was only tolerated at the Academy because of his brilliance.