Pythagoras’s theorem leads to one of the best of understood equations in mathematics:
x2 + y2 = z2.
There are many whole number solutions to this equation,
e.g., 32 + 42 = 52.
In the seventeenth century the French mathematician Pierre de Fermat set a challenge for future generations of mathematicians – prove that there are no whole number solutions for the following closely related family of equations:
x3 + y3 = z3,
x4 + y4 = z4,
x5 + y5 = z5,
x6 + y6 = z6,
Although these equations appear similar to Pythagoras’s equation, Fermat’s Last Theorem claims that these equations have no solutions. The difficulty in proving that this is the case revolves around the fact that there are an infinite number of equations, and an infinite number of possible values for x, y, and z, and hence the proof has to check that there does not exist a solution within this infinity of infinities. Nonetheless Fermat claimed he had a proof. The proof was never written down and ever since the challenge has been to rediscover the proof of Fermat’s Last Theorem.
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. Mathematicians had merely succeeded in showing that there are no solutions to the following equations:
x3 + y3 = z3,
x4 + y4 = z4.
An infinity of other equations remained, and mathematicians still had to demonstrate that none of these had any solutions. There was no progress until a young French woman reinvigorated 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.
Sophie Germain was born on 1 April 1776 the daughter of a merchant, Ambroise-Francois 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. Had she been born into high society, then her study of mathematics might have been more acceptable. Although aristocratic women 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 text books 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 time 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-Étienne 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. But this sudden interest in such an unfeminine subject worried her parents and they tried desperately to deter her. A friend of the family Count Guglielmo Libri-Carrucci dalla Sommaja wrote how Sophie’s father confiscated her candles and clothes and removed any heating in order to discourage her.
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 straight-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 claimed 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 undoubtedly she was also immensely determined. Eventually her parents relented and gave Sophie their blessing.
Germain never married and throughout her career her father funded her research and supported her efforts to break into the community of mathematicians. For many years this was the only encouragement she received because there were no mathematicians in the family who could introduce her to the latest ideas and her tutors refused to take her seriously.
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 hence 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 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 marvelously ingenious but they showed a remarkable transformation in a student who had previously been notorious for his abysmal mathematical skills. 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 widely acknowledged as being the most brilliant mathematician 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. Historians have suspect that 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.
Germain had adopted a new approach to the problem which was far more general than previous strategies. Her immediate goal was not to prove that one particular equation had no solutions, but to say something about several equations. In her letter to Gauss she outlined a calculation which focused those equations in which n is equal to a particular type of prime number.
Prime numbers are those numbers which have no divisors. For example, 11 is a prime number because 11 has no divisors, i.e. nothing will divide into 11 without leaving a remainder (except for 11 and 1). On the other hand, 12 is not a prime number because several numbers will divide into 12, i.e., 2, 3, 4, and 6. Germain was interested in those primes numbers (p) such double the prime add one (2 x p+ 1) is also a prime number. Germain’s list of primes include 5, because 11 (2 x 5 +1) is also prime, but it does not include 13, because 27 (2 x 13 +1) is not prime.
For values of n equal to these Germain primes, she could 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 must be a multiple of n. This 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, therefore showing that for that particular value of n there could be no solutions.
In 1825 her method claimed its first complete success thanks to Johann Peter 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 a brilliant 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, the work of Sophie Germain.
Fourteen years later the French made another breakthrough. Gabriel Lamé made some further ingenious adaptations 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, asking that he guarantee Gauss’s safety. The general was not a scientist but even he was aware of the world’s greatest mathematician, and, as requested, he took special care of Gauss, 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 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 familiarize 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 honored it.
Sophie Germain’s correspondence with Carl Gauss inspired much of her subsequent 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, others were to build on her work. She had offered hope that those equations in which n equals a Germain prime could be tackled, however the remaining values of n remained intractable.
After Fermat, Germain embarked 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 was to lay 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 honor 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 toward establishing the theory of the elasticity of metals – Sophie Germain. Was she excluded from this list for the same reason that Agnesi was ineligible to 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 toward 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