Sophie Germain Dionna Bidny

Sophie Germain’s Life

The term revolutionary can apply to a great array of individuals. Some revolutionaries ﬁght social standards with weapons, some with words, and some with thoughts. During not only her lifetime but well beyond, Sophie Germain was, in more than one sense, a revolutionary. Her life and actions set a new standard for not only mathematics but also for women, emerging from a changing time reﬂected in the French Revolution itself.

Sophie Germain was born on April 1, 1776, the second of three daughters of a middle class family. When Germain was only 13 years old, the French Revolution erupted around her with the fall of the Bastille. Forced to stay indoors, Germain took refuge in her father’s library, where she poured over books on mathematics, Latin, and Greek. She read works by Euler, Newton, and many other scholars, and was determined to follow the paths they paved and delve even deeper into the world of mathematics.

Germain’s parents, however, were unsupportive of her new scholarly interests that were so atypical and socially unacceptable for women of the time, especially those of Germain’s age. It is said by her ﬁrst biographer, Italian mathematician Guglielmo Libri, that Germain was forced to study in the cold library by candlelight at night because of her family’s attempts to thwart her budding love of mathematics, language, and philosophy at any other time. Undeterred, it is evident that Germain persevered and did all she could to purse her studies in the subjects that sparked such an interest in her. Her parents, eventually noting her undying zeal, eventually allowed her to continue her education.

As Germain diligently studied the works of Bezout, Cousin, and many other scholars in her father’s library, the Revolution around her swelled. As a direct result of the Revolution, the Ecole Polytechnique opened successfully in Paris in the year 1794, publishing its first journal one year later. The academy strove to provide an education strong in mathematical and scientific learning to enlighten the middle class and especially to compensate for the death of many engineers and officials during the Revolution. With world-renowned professors as dedicated to research as they were to classroom teaching, the Ecole quickly gained momentum in popularity and prestige. Unfortunately for Germain, however, the Ecole did not admit women as students until much later. Undeterred as always, 18-year old Germain somehow obtained books and lecture notes, which she ever-enthusiastically studied, from students attending the Ecole. Germain especially focused on the works of mathematician Joseph-Louis Lagrange, who taught at the institution. At the Ecole, it was customary for instructors to invite students to write observations or analyses of the works they had been taught by the end of the term. Cleverly, Germain submitted her work on Lagrange’s study of mathematical analysis under a pseudonym of a student already attending the Ecole (Antoine-August LeBlanc). Naturally, Lagrange was extremely impressed by Germain’s review, and even more impressed and pleasantly surprised when he discovered here true identity after he sought out a meeting with her.

It’s important to note that at eighteen, Germain had already achieved the educational level of an undergraduate student, enough to impress a top mathematician of the time, simply through her own dedication and self- study. The fact that all of her mathematical prowess came to Germain not only without assistance, but actually during a time when others attempted to suppress her interest, is nowhere short of astounding. Many previous successful female mathematicians started their careers under the mentorship and encouragement of an already established mathematician. Germain, however, had only her incredible passion and insatiable thirst for knowledge as her sole motivator until much later in life.

After their initial acquaintance, Lagrange and Germain continued their scholarly correspondence. This correspondence gained Germain exposure to much of Paris’ notable mathematical circle. She began to collaborate with other renowned thinkers, and engage in new ideas surrounding intricate mathematical, philosophical, and literary concepts. A newfound interest in number theory prompted Germain to contact a prominent mathematician of the time, Legendre, whose book Th´eoriedes Nombres focused on this idea precisely. Their correspondence was so in-depth, that Legendre in fact credited Germain with more developed concepts that he included in a later edition of Th´eorie.Despite now being a more elite member of the Parisian mathematical circle, it is evident that Germain’s proactiveness and determination for learning never waned, as she continued to stretch her limits and gain mentorship and insight from anyone with whom she could communicate.

Germain similarly began correspondence with Gauss one again under the pseudonym LeBlanc (the real LeBlanc having died by now) after reading on of Gauss’ books. Germain included her work on Number Theory and her preliminary work on Fermats Last Theorem. Gauss, naturally, was impressed by the thoroughness of Monsieur LeBlanc’s thought processes. When Gauss learned the true identity of is intelligent young correspondent, he was surprised but full of encouragement and praise, as is evident in a statement he made after discovering the true Germain:

But how can I describe my astonishment and admiration on seeing my esteemed correspondent Monsieur LeBlanc metamorphosed into this celebrated person, yielding a copy so brilliant it is hard to believe? The taste for the abstract sciences in general and, above all, for the mysteries of numbers, is very rare: this is not surprising, since the charms of this sublime science in all their beauty reveal themselves only to those who have the courage to fathom them. But when a woman, because of her sex, our customs and prejudices, encounters

2 infinitely more obstacles than men, in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtless has the most noble courage, extraordinary talent, and superior genius. Nothing could prove to me in a more flattering and less equivocal way that the attractions of that science, which have added so much joy to my life, are not chimerical, than the favor with which you have honored it. The scientific notes with which your letters are so richly filled have given me a thousand pleasures. I have studied them with attention and I admire the ease with which you penetrate all branches of arithmetic, and the wisdom with which you generalize and perfect.

At this point, Sophie Germain embarked on a number of notable mathematical journeys (the details of which will be covered in a later section). Among these was a major portion of a proof of the vibrational and elastic waveforms observed on a surface by German physicist Ernst Chaladni in 1808. By demonstrating nearly to completion the underlying mathematics of the demonstrated phenomenon, Germain eventually won a contest by the French Academy of Sciences to develop such a proof. This award was highly influential in Germain’s life, as it propelled her to the ranks of a higher circle of mathematicians. After winning the award, Germain continued to improve her work and became ultimately very influential in the field of elasticity.

In addition to her work in elacticiy, Germain dedicated extensive work towards developing a proof of Fermat’s Last Theorem. While Germain never completed the proof, Germain is credited for developing the ﬁrst proof that generalizes the theorem and makes it applicable for a range of exponents. In addition, manuscripts of Germain that were lost and later recovered show that her work extended far past this portion of the proof and delved deep;y into a method for proving the theorem in its entirety.

Germain continued to rise as a prominent mathematician of her time. She was the ﬁrst single woman to attend the Academy of Sciences session after an invitation by the Insitute of France, and was conintually praised by the Institute and her mathematical peers. By now, Gauss had convinced the University of Gottingen to award Germain with an honorary degree.

However, before she could receive the honorary degree, Germain died to brest cancer on June 27th, 1831. She was 55 years old. While premature, her death did nothing to minimize the revolutionary life she had led. Germain persevered through political upheaval, social prejudice, and lack of formal education to achieve more than seems possible. She left an extremely signiﬁcant imprint on the face of mathematics, and her works and memory continue to inﬂuence many to this day.

3 Sophie Germain’s Mathematical Works

FERMAT’S LAST THEOREM:

As has been mentioned, one of Sophie Germain’s most prominent achievements was her work with Fermat’s Last Theorem. The origin of the theorem stems back to a note scrawled by Pierre de Fermat in the margin of a book in the 1630s:

It is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or in general any power higher than the second into two powers of like degree; I have discovered a truly remarkable proof which this margin is too small to contain

The proof that went unwritten in the margin was, for all we know, never written down at all by Fermat, but rather continued to dog at the minds of mathematicians for centuries.

In mathematical notation, the theorem Fermat developed is as follows: xn + yn = zn has no positive integer solutions for x, y, z when n > 2

While this theorem wasn’t fully proved until Andrew Wile’s 1995 publication, Sophia Germain played an essential role in ﬁrst creating an overarching plan for how to tackle a majority of the proof.

Previously, Fermat had proven by his method of inﬁnite decent that for a right triangle with rational sides, the area cannot be a square. In other words, x4 + y4 cannot equal z2 or z4 because the area would be a product of perfect squares: (1/2)(x2)(y2). This of course proves Fermat’s Last theorem only for n = 4.

This argument can of course be applied to every integer exponent that is a multiple of 4. If the exponent is a multiple of 4, say 4p, the following equation could not be satisﬁed either: x4p + y4p = z4p

4 In 1770, Euler published a proof for Fermat’s Last Theorem for n=3. While the proof was later found to be incomplete, Gauss corrected the proof, which published after his death. Therefore, when Germain set out on her own journey for the proof of the theorem, she had only knowledge of these partial proofs to work oﬀ of. She shared her interested in proving the theorem ﬁrst in a letter to Gauss:

I add to this art some other considerations which relate to the famous equation of Fermat xn + yn = zn whose impossibility in integers has still only been proved for n = 3 and n = 4; I think I have been able to prove it for n = p-1, p being a prime number of the form 8k+7. I shall take the liberty of submitting this attempt to your judgment, persuaded that you will not disdain to help with your advice an enthusiastic amateur in the science which you have cultivated with such brilliant success.

Germain never completed the proof of Fermat’s Last Theorem, but her advances were instrumental to future work and ultimate success in the area. In the end, she became the ﬁrst person to develop a generalized proof of any kind for the theorem.

After her ﬁrst letter to Gauss about Fermat’s Last Theorem, Germain abandoned the work for a period of time. Years later, she returned to the subject in another letter to Gauss, stating how the challenge of the theorem had so often tormented her. It was at this point that Germain developed a Grand Plan of how to tackle the proof.

In order to create a proof that could be generalized to many integer exponents, Germain aimed to provd that for every odd prime exponent P, there exist an inﬁnite number of auxiliary primes of the form 2Np + 1 so that the set of xp mod (2Np + 1) contains no consecutive integers. In other words, when a power xp is divided by (2Np + 1), the set of remainders contain no consecutive integers. (Possible remainders for any m mod n are of course from the set of consecutive integers 1-n). With this information, Germain realized that if a solution to xp + yP = zp existed, then the auxiliary prime (2Np + 1) would have to divide either x, y, or z.. She performed calculations to demonstrate this for all primes less than 100 and for the axiliary primes (2Np + 1) for 1 ≤ N ≤ 10

To further understand Germain’s arithmetic, let us demonstrate such an example with p = 5 and N = 1. This would mean that we would look at all possible x5 mod (2(1)(5) + 1) = 11. Of course, modulus results would fall between 1 and 11. These values are as follows:

5 (15, 25, 35, 45, 55, 65, 75, 85, 95, 105) mod 11

= (1, 32, 243, 1024, 3125, 7776, 16807, 32769, 59049, 100000) mod 11

= (1, 10, 1, 1, 1, 10, 10, 10, 1, 10)

= (1, 10)

1 and 10 are not consecutive, so therefore 11 is an auxiliary prime that works for p = 5 according to Germain. Similarly, this can be attempted for all values for 1 ≤ N ≤ 10 that result in a prime when applied to (2Np+1). There are a number of ways for an N value to fail the conditions described by Germain. For example, N = 2 involves mod 21, which is not a prime number and thus fails to create an auxiliary prime to use in the modular arithmetic.

Other sets involve a prime auxiliary number, such as N = 3 involving mod 31, but result in power residues that fail to satisfy the non-consecutive reside condition. Once again for N = 3, the 5th power residues mod 31 result in the set (1, 5, 6, 25, 26, 30) which contains consecutive integers. On the other hand, N values such as N = 4 satisfy the non-consecutive power residue condition. N = 4 has 5th power residues (1, 3, 9, 14, 27, 32, 38, 40) which contains no consecutive elements.

Following this method for all 1 ≤ N ≤ 10 and p = 5, it can be shown that the N values of 1, 4, 7, and 10, which correspond to auxiliary prime (2Np + 1) values of 11, 41,71, and 101 respectively, satisfy the non-consecutive power residue condition. Thus, according to Germain’s statement, for x5 + y5 = z5 to be true, each of the above-mentioned auxiliary primes would have to divide either x, y, or z. In other words, x, y, and z would have to each be multiples of at least one such auxiliary prime.

Had Germain been able to prove that and infinite number of such auxiliary primes 2Np + 1 exist for every odd prime exponent p, her proof to Fermat’s Last Theorem would have been complete. For, if infinite auxiliary primes were the case, each of those primes would have to divide at least one of x, y, or z. If we create three subsets of auxiliary primes with one subset containing the auxiliary primes that divide x, one containing those that divide y, and one that contains those that divide z, at least one of those subsets must be infinite. This, naturally, would mean that at least one of x, y,, or z must be divisible by an infinite number of primes, which of course is impossible. Thus, Fermat’s Last Theorem would have no solutions to that prime exponent p.

But as the case was, Germain was not able to fully generalize her theorem to any odd prime exponent p, as she was unable to prove the fact that an infinite number of auxiliary primes 2Np + 1 exist for every p. In fact, it was proven that the number of auxiliary primes for a given p is actually finite. Germain even later proved herself that there exist only two auxiliary primes (7 and 13) that work for p = 3. However, Germain’s work is still a milestone in the history of Fermat’s Last Theorem, as it marked the first plan for a proof that works for a general and infinite p rather than a case by case basis.

When Germain realized that she was unable to complete her grand plan to prove Fermat’s Last Theorem, she declared her defeat in a letter to Gauss. However, she also included in the letter some of her work that demonstrated that solutions to Fermat’s equation would have to involve very large integers. She developed the following key theorem:

For an odd prime p, if the equation xp + yp = zp is satisﬁed in integers, then one of the numbers x + y, z − x, or z − y must be divisible by p2p−1 and by the p − th power of all primes of the form 2Np − 1 which satisfy the two conditions:

1. There are not two consecutive non-zero p − th power residues (mod 2Np + 1)

2. p is not a p − th power residue (mod 2Np + 1)

6 In addition to the ﬁrst condition that Germain had in place for her original grand plan, she now added the second condition that states that p itself cannot be a solution for the pth power residue mod the auxiliary prime 2Np + 1. In other words, the set of auxiliary primes that satisfy condition 1 cannot contain p itself. For example, the set of auxiliary primes we calculated for p = 5 does not contain 5, and thus fulﬁlls both conditions. Germain’s claim in her key theorem would therefore suggest that for any solution to x5 + y5 = z5, then the numbers x + y, z − x, or z − y must be divisible by 59 (which is p2p−1) as well as by 115, 415, 715, and 1015 (which are the auxiliary primes of p = 5 raised to the 5th power). In other words, the numbers x + y, z − x, or z − y must be divisible by the product

(59)(115)(415)(715)(1015)

=691,053,006,763,356,095,514,121,490,614,455,078,125

Thus, one of either x, y, or z would thus have to have a value that is at least half of this massive number. When describing this key theorem to Gauss, Germain stated that this value would be one whose size frightens the imagination However, Germain’s proof of her key theorem was incomplete and not accurate, something she herself was aware of. In a later attempt to correct her mistakes, she was unsuccessful. Nonetheless, when compiling her work she was still able to deﬁnitely conclude the following:

If the conditions of the key theorem are satisﬁed for an auxiliary prime 2Np + 1, then one of x, y, or z is divisible by p2.

Germain’s work led to a greater understanding of Fermat’s Last Theorem and its being broken down into two cases:

Case I: xp + yp = zp has no integer solutions for which x, y and z are relatively prime to p, i.e, in which none of x, y, and z are divisible by p

Case II: xp + yp = zp has no integer solutions for which one and only one of the three numbers is divisible by p

As a summary of her work, Sophie Germain compiled her studies into a proof that is today known as Sophie Germain’s Theorem that proved an instrumental portion of the completion of the proof for Fermat’s Last Theorem.

Sophie Germain’s Theorem:

Let p be and odd prime. If there is an auxiliary prime θ satisfying the two conditions:

1. xp + yp + zp = 0modθ implies that x = 0modθ, y = 0modθ, or z = 0modθ, and

2. xp = pmodθ is imposible for any value of x, then Case I of Fermat’s Last Theorem is true for p.

These two conditions are the same conditions as can be found in Germains key theorem.

Germain continued her work on Fermat’s Last Theorem throughout most of her life, and was given much credit for her work that Lagrange included in one of his major papers. Eventually, Lagrange and Germain together were able to prove that all prime numbers less than 197 satisfy Case I of Fermat’s Last Theorem.

Finally, any prime number p such that 2p + 1 is now known as a Sophie Gemain prime, and actually has applications in number theory and cryptology. Sophie Germain’s work in Fermat’s Last Theorem is some of her most inﬂuential in the realm of number theory.

7 ELACTICITY:

Ernst Chladni, German physicist and musician, presented his experiments at the Paris Academey of Sciences in 1808 and greatly sparked Germains interest. Chladni’s experiments involved drawing a violin bow over a flat metal surface lightly dusted with sand. When the plate reached resonance frequency with the sound created by the bow, the sand on the plate would form patterns centering on the nodal points of the perpetrating sound wave along the flat plate. Changes in sonar frequency would naturally affect the sound wave and thus affect the pattern of the sand.

When Napoleon heard of Chladni’s experiments, he invited him to present at the Academy. Following the presentation, the Institut de France held a competition for individuals to mathematically prove the phenomenon that Chladni experimentally proved. Germain set eagerly to work. Most mathematicians did not even bother submitting papers as they assumed that the mathematics of the time were insuﬃcient to describe Chladni’s experiment. This resulted in Germain being the only applicant to the competition with her preliminary explanation, but the judges did not consider her proof to be suﬃcient and thorough enough.

The contest was then extended, and Germain continued to modify and improve her proof. This time, her work won honorable mention, as she still was not able to develop a basis of fundamentals for her mathematical explanation. Finally, Germain submitted her paper Recherches sur la th´eoriedes surfaces ´elastique (Memoir on the Vibrations of Elastic Plates) in 1816, and won the prize.

Her ﬁnal equation for the vibration was:

2 δ4z δ4z δ4z δ2z N ( δz4 + δx2δy2 + δy4 ) + δt2 = 0

Where N 2 is a constant.

8 Historical events that marked Sophie Germain’s life.

Naturally, the major event surrounding Germain’s life was the French Revolution itself. As a decently wealthy middle class family, the Germains were not dreadfully threatened by the Revolution when it broke out in 1789. However, the family had to remain mostly indoors due to the looming threat of political extremism as the Revolution proceeded.

The years 1789-1792 were marked by the eruption of the Revolution, subsequent riots against all royal mo- nopolies, abolition of titles of nobility, and resistance against all religious orders and communities. Ultimately, King Louis XVI was imprisoned after rioters stormed the palace. A year later, the King was executed, and France declared war on Brittan and Holland.

The revolution then continued with rivalry no longer between the middle class and royals, but rather between the working and middle class. The politics and seat of power grew more out of control, and while execution of royals such as Marie Antoinette continued, thousands of others were guillotined, many of whom were in fact poor.

As a ﬁnal phase in the Revolution, the French people began retaliating against the extremism until ﬁnally Napoleon Bonaparte took control of the French army and the Directorate of the Revolution was forced to resigns in 1799. Napoleon then took hold of the position of First Consul, which essentially gave him a dictator’s role. In 1804, he became Emperor of France.

Sophie Germain lived to see not only the rise and fall of the Revolution, but also the rise and fall of Napoleon himself. After a failed invasion of Russia in 1812, Napolion was exiled to the island of Elba. After brieﬂy returning to France in 1815, he was once again crushed in the Battle of Waterloo. Napoleon was again exiled, this time to the remote St. Helena, and died there at the age 51.

9 Signiﬁcant historical events around the world during Sophie Germain’s life

Throughout Sophie Germain’s life, nations across the globe were embroiled in their own changing times. Most notable, of course, was the revolution occurring across the Atlantic Ocean: The American Revolution, which began just a year before Germain was born in Paris. After the signing of the Declaration of Independence, British troops invaded and occupied a great number of American states. Eventully, France and America join forces by signing the French Alliance in 1778. Ultimately, this led to the signing of the Treaty of Paris by the United States and Brittan, which ended the revolution. In 1787, the United States Constitution was signed.

In Europe, history of the late 18th century and early 19th century often centered around Napoleon’s conquests. In 1796, he invaded Austria, and extends his invasion to Russia and Egypt two years later. Finally, in 1800, Napoleon conquered Italy. A year after the U.S. completed the Louisiana Purchase that doubled the size of its land, Haiti declared independence from France in 1804, thus establishing its place as the ﬁrst black nation to gain independence from European rule. Despite this loss of colonial rule, Napoleon continued his conquest and occupied Spain. Shortly after, Napoleon rashly invaded Russia and was forced to retreat. This defeat ultimately led to Napoleons defeat by the Allied Nations in 1814.

10 Signiﬁcant mathematical progress during Sophie Germain’s lifetime

France was a prominent source of mathematical development toward the end of the 18th and start of the 19th centuries. In addition to Germain’s own work, other French mathematicians (many of whom corresponded with Germain) developed many breakthroughs during this time.

Joseph Louis Lagrange, previously mentioned as one of the first prominent mathematicians to discover Ger- main’s brilliance, developed a strong foundation for differential equations and number theory. In addition, Lagrange also developed what is now known as Lagrange’s Mean Value Theorem. This theorem states that given a differentiable section of a curve, there will be at least one point on that curve whose derivative is equal to the mean derivative of that curve section. Lagrange was also a prominent physicist, and developed a remarkably comprehensive mathematical summary of classical mechanics based on Newton’s work.

Adrien-Marie Legendre, who also worked closely with Germain, was notable in his work in number theory, mathematical analysis, and statistics. His work includes the prime number theorem, which describes the distribution of prime numbers within positive integers. However, much of Legendre’s achievements were not perfected nor presented to the public eye until done so in later years by Gauss.

Toward the beginning of the 19th century, Frenchman Joseph Fourier published his work on infinite sums in which the terms are trigonometric functions. To this day, periodic functions that can be written in the form of a sum of an infinite series of sines or cosines are known as Fourier series. In Germany around this time, Carl Friedrich Gauss was studying at the University of Gottingen. Gauss, who also corresponded with Germain, would go on to become an incredibly influential mathematician, publishing works far ahead of their time in abstract algebra, geometry, vector math, and many other arenas.

11 The French Revolution and mathematical development

During Germain’s time, history had a massive impact on the development of mathematical progress. As he rose to power, Napoleon placed signiﬁcant stress on the importance of mathematical education and its practicality. Many engineers and individuals with mathematical knowledge were killed during the revolution, and renewed eﬀort was placed into developing schools and educational systems with a focus on mathematics and technology.

For the ﬁrst time in French history, mathematics and engineering was taught to those with talent and dedication, not simply those with social standing. With the collapse of nobility and titles during the Revolution, mathematical education, prowess, and collaboration was erupting on a greater scale than ever before. It re- mains apparent that Germain, fortunate enough to have been born into a world of such quickly developing intellect, played an important role of contributing to this development. Her passion, focus, and creativity all show her to be one of the mots prominent thinkers of France’s mathematical revolution.

12 References