Until lately, Sophie Germain has been a mathematician hidden in the shadows of other great mathematicians. Her works, mainly on elasticity and Fermats last theorem, were hardly ever heard of until Andrew Wiles proof of Fermats last theorem was published. It was then that everyone wanted to look back into the history of this long elusive theorem. It was then that they found Sophie Germain. Why has she been so unknown until recently? Many factors may be the answer: she was female, she was a member of the Bourgeoisie and not the Aristocracy, she was self taught, but never really got the hang of some key mathematical ideas, and maybe even because her work opened a pathway for other men to work on what she had started. These reasons alone, however, should be reason enough to study her and learn about her contributions to the fields of mathematics and science.
Marie-Sophie Germain was born April 1, 1776 in Paris, France, in the era of the French Revolution. Her father, Ambroise-François, was a wealthy silk merchant and a member of the Bourgeoisie. He was also a representative in the États-Généraux (Gray 47) and became a director of the Bank of France (Swift 1). Little is known about her mother except for her name, Marie-Madeleine Gruguelin. Germain never married and lived at home all of her life. Even through the Revolution, her father remained prosperous and supported her financially (Gray 47).
Germains home was a place for political and philosophical discussions, especially pertaining to liberal reforms (Marie-Sophie Germain 1). The Revolution and the political upheavals that resulted were partially the reason for Germains interest in math. When the revolution started, she locked herself away in her fathers library and read. She was always known to be a reclusive character and perhaps without this trait she would never have found the inspiration to become a great mathematician. It was in her fathers library that she discovered the legend of Archimedes death in Jean-Étienne Montuclas History of Mathematics (Maths Hidden Woman 2,3). The story says, "during the invasion of his city by the Romans, Archimedes was so engrossed in the study of a geometric figure that he failed to respond to the questioning of a Roman soldier. As a result he was speared to death." This inspired Germain so much that it was from this point on that she knew she wanted to be a mathematician (Swift 1 and others).
Also, in this library, Germain took her education into her own hands. She began reading about and teaching herself mathematics. Her parents, however, thought that this was an inappropriate interest for a female and fought with her to discourage it in the beginning. Germain started studying while her parents slept. As a result, she ended up having to study wrapped in blankets by the light of smuggled candles because her parents had taken away her fire, her light, and even her clothes in an attempt to force her away from her books. These types of beliefs were common in the middle class families of the nineteenth century (Swift 1 and others). Even in the aristocracy, women were only supposed to have enough knowledge of math and science as to make pleasant conversation about the subjects. Because of this, Francesco Algarotti wrote Sir Isaac Newtons Philosophy Explaind for the Use of Ladies. Algarotti believed that women were only interested in romance and so his explanations of various laws of physics were in the guise of flirtatious dialogues between a Marquise and her interlocutor. The Marquises explanation of the inverse square law of gravitational attraction is that, "I cannot help but thinking that this proportion in the squares in the distance of places is observed even in love. Thus after eight days absence, love becomes sixty-four times less than it was the first day" (Maths Hidden Woman 2). Needless to say, this was not the type of book that interested Germain.
Studying at night, Germain was able to teach herself Latin and Greek. She read Newton, Euler (Marie-Sophie Germain 1), and Cousin. It was so cold on some nights that the ink in her inkwell froze. Eventually her parents gave up the fight and gave her their blessings. In truth, her mother had secretly been supporting her the whole way, bringing her books and candles (Gray 48). This self-teaching, however, was not enough for Germain. Unfortunately, there was very little secondary education in mathematics at the time. The only school that did make it available was the Ecole Polytechnique, founded in 1795 as part of educational reforms after the revolution. This liberal institute of higher learning, however, barred women ( 192). Two innovations that the institute did make were providing lecture notes to all that asked, and the practice of having students submit written observations. Germain used these two innovations and learned from many of the prominent mathematicians of the day. She was especially interested in a course on analysis by one of the first professors at the institute, Joseph Louis Lagrange (Gray 48). Germain submitted a report on analysis to Lagrange using the name of an acquaintance registered as a student at the school, Antoine-August Le Blanc, or better known as Monsieur Le Blanc, because she felt her answers would not be accepted if it was known that the author was female ( 192). Lagrange became curious because of the transformation of a student that was once notorious for his abysmal math skills to one who produced solutions that were marvelously ingenious. Lagrange began to search out the student and requested a meeting with him. He was surprised and pleased to find that the author was a woman. Lagrange became Germains mentor and friend (Maths Hidden Woman 4). Lagrange introduced her to the circle of scientists and mathematicians that she was never before able to access. Both her gender and social status had been a hindrance until this time. Germains education, in the end, was overall haphazard and not well pieced together.
Germain also started correspondences with other great mathematicians. When Adrien Marie Legendre published his Essai sur le Théorie des Nombres in 1798, she worked on it intensely and began a correspondence with him. The Legendre-Germain correspondences became a collaboration and Legendre published many of Germains discoveries in a supplement to his second edition of the book (Marie-Sophie Germain 1). Legendre later helped her on her own works pertaining to elasticity and number theory. Her best-known correspondences, however, were with Carl Friedrich Gauss. After reading and studying his 1801 Disquisitiones Arithmeticae, she wrote to him about her own works and problems on number theory. She again used the pseudonym M. Le Blanc because she feared being ignored because of her sex. Gauss gave her number theory proofs high marks ( 193). Eventually, Germain did reveal to Gauss her real identity. In 1806, the French occupied Gausss hometown of Braunchweig. She feared that he would suffer the same fate as Archimedes and decided that she must intervene. Germain contacted a family friend in the French forces who sent a battalion chief to check on Gauss. The chief told Gauss that this intervention was due to a Mlle. Germain. Gauss confessed that he did not know a Mlle. Germain, and so she sent him a letter to clear up the confusion. Germain explained to Gauss of her fear of ridicule because of her sex and the disrepute attached to the femme-savantes of the time. Gauss praised her even more highly after this discovery (Gray 49). Gauss responded:
But how can I describe my astonishment and admiration on seeing my esteemed correspondent Monsieur Le Blanc metamorphosed into this celebrated person, yielding a copy so brilliant it is hard to believe? But when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men, in familiarizing herself with their knotty problems, yet overcomes these fetters and penetrates that which is most hidden, she doubtlessly has the most noble courage, extraordinary talent, and superior genius ( 194)
Gauss also credits Germain for encouraging his return to number theory (Gray 49).
Germainis best known for her studies on elasticity and number theory, in particular, Fermats Last Theorem. Germains interest in elasticity came about in 1808 when a German physicist, Ernst F. F. Chladni, gave lectures in Paris about vibrating plates. Chladni produces curious patterns on small glass plates covered with sand and played by using a bow. The sand moved until it reached nodes and the array of patterns resulting from the "playing" of different notes. This was the first scientific visualization of two-dimensional harmonic motion (Sophie Germain 1). In 1809 the Institut de France, of which Napolean was the president, announced a competition with a prize set at a kilogram of gold (Dauben 65). The challenge was to, "formulate a mathematical theory of elastic surfaces and indicate just how it agrees with empirical evidence," and a deadline of two years was set (Marie-Sophie Germain). Germain wrote the first paper on the subject and submitted it in 1811. In fact, she was the only applicant for the award. It was conjectured to other potential competitors that the problem would be extremely difficult and would require a new type of analysis. Lagrange said that the type of math needed to achieve this goal at the time was not yet available. Germain, herself, did not have the grasp on double integrals that was necessary for this type of work. Germain did not win the award in this year. Her hypothesis was not derived from principles of physics and she did not have the required understanding of analysis and variation. The judges found many errors with her paper in the calculations, but her ideas gave scientists and mathematicians a new way to look at the problem. Her paper did contain a valid assumption of the nature of the problem. Lagrange fixed her errors in her equation and, since the deadline for the competition was extended two years, she submitted another paper. She, again, was the only entrant. Her new calculations showed some of the patterns, but she could not give a satisfactory derivation of Lagranges equation from physical principles. She was awarded a prize of Honorable Mention for her connection of the physical, experimental data and the correct differential equation. The Institut extended the award one more time until October 1815. She entered another memoir, working this time totally in isolation. Yet again the sole entry, her memoir won the prix extraordinaire in 1816. The judges said it was interesting in intent but hardly better than any of her other papers (Dauben 67). She did not appear at the award ceremony because she thought that the judges did not fully appreciate her work and she wanted to avoid a public spectacle. She said that the scientific community did not show the respect that seemed due her. Her chief rival on the matter even said, "Although it was Germain who first attempted to solve a difficult problem, when others of more training, ability, and contact built upon her work, and elasticity became an important scientific topic, she was closed out. Women were simply not taken seriously" (Marie-Sophie Germain 2).
Germain most famous work, however, is her efforts at proving Fermats Last theorem. Fermat stated that it is not possible to find, for a given integer n>2, non zero integers x, y, and z such that:
This theorem had already been proven for n=3 by Euler , for n=4 by Fermat, and for n=5 by Legendre. Fermat, himself, stated, "I have discovered a truly remarkable proof which this margin is too small to contain." Unfortunately, he didnt leave it for anyone to find. Although this theorem was proven by Wiles in 1995, his proof sparked a new interest in the mathematics of Sophie Germain. Germain was the first person to come up with a more generalized way to prove Fermats Last Theorem. Before her, the only proofs had been for specific cases of n. First, Germain knew that the only numbers that were necessary to look at were odd primes, because all other cases of n could be broken up into these. Because of this result, it sufficed to prove that the equation had no solutions for all prime exponents p_3, where x, y, and z are pairwise coprime. Then it was split up into two cases:
Case 1 -- the exponent p does not divide x, y, or z;
Case 2 the exponent p does divide x, y, or z.
Germain worked on Case 1 of the theorem (Granville and Monagan 329). She restated FLT in the form:
where the equation has no solutions not divisible by θ. Mod θ means that if you divide a number by θ, then the remainder would be that number mod θ. An example of this would be that 2 is 16 mod 7. This means if 16 is divided by 7, 2 is the remainder. If a number is divided by θ evenly, then the number mod θ is zero. Another example would be 14 mod 7 is zero because 7 divides into 14 evenly. The form of the equation is possible because, since p is always odd, a negative number can be put in the place of z and the equation will still hold true. Her theorem is If p is an odd prime, and if there exists an auxiliary prime θ with the properties that
1. p is not a pth power modulo θ (or p_lp mod θ), and
2. the equation r ≡ r+1 modulo θ cannot be satisfied for any two pth power residues,
then Case 1 of FLT is true for p.
If this prime is found, then there does not exist an integer solution not divisible by p, or in mod θ, not zero. Germain found that this would work for primes p and θ=2p+1. These primes are called twin primes, or Sophie Germain primes, and are still studied today. People are rushing to find larger Sophie Germain primes and some are even looking for special types of these primes, such a palindromic Sophie Germain primes. A palindromic number reads the same backwards and forwards. The smallest palindromic SG prime consists of 23 digits. It is p=19091918181818181919091. This would make 2p+1=38183836363636363838183, another palindromic prime. One step farther even produces yet another palindromic prime in the form of 2(2p+1)+1= 76367672727272727676367. The largest known set like this contains 72 digits in each prime (Dubner).
An example of a p and a θ=2p+1 would be p=3 and θ=2(3)+1=7. This would make the equation x3+y3≡z3 (mod 7). It can be shown that 3_l3 (mod 7), in fact, anything raised to the third power mod 7 is either 1, 6, or 0. Since the hypothesis states that we dont want numbers divisible by θ, or 7, this means we can throw out the zeros. Now we can just look at why we will always get 1 or 6. Any number raised to the third power (that is not 0 mod7) can be written as (7n+a)3 where n is an integer and a is an integer between 1 and 6 inclusive.
(7n+1)3 = 343 n3 +147 n2 + 21 n + 1. Since 343 n3, 147 n2, and 21 n are all divisible by 7 because they are all products of 7n and another number, they will all be 0 mod7 and drop out of the solution. This means the only important number is the last term, or a3 because it is not a product of 7n, and therefore will not reduce to zero. Hence, (7n+1)3=1.
Again, the only necessary part of the solution to look at will be a3 because all of the other terms contain a 7n. Since 23=8, which is 1 mod7, which goes along with the conjecture that the residues, or remainders, will be either 1 or 6, since 0 and 7 will just provide another multiple of 7.
For a=3: 33=27, or 6 mod7.
For a=4: 43=64, or 1 mod 7.
For a=5: 53=125, or 6 mod 7.
For a=6: 63=216, or 6 mod 7.
Since all integers not divisible by zero can be written in one of these forms, it follows that 3_l3 mod 7. It also follows that since 1 and six are non-consecutive numbers, the second stipulation holds. In the same way, it can be shown that for p=5 and θ=2(5)+1=11, the residue is always 1 or 10. This means that 5_l5 mod11 and the residues are non-consecutive.
In the case that p is prime, but 2p+1 is not, it can be shown that these numbers invalidate both conditions of the hypothesis. Such an example of this would be p=7 and θ=15. This would mean that a would range from 1 to 14. A list of a7 would look like: 1, 8, 12, 4, 5, 6, 13, 1, 8, 12, 4, 5, 6, 13, 2, 9, 10, 11, 3, 7, 14. It is obvious to see that some of these numbers are consecutive and 137 even equals 7 mod 15. These results can be generalized to the form (θn+a)p modθ where n is an element of the integers and a is an integer between one and (θ-1) inclusive. Since we are looking at the final term of the expansion, ap, we can say that it will be (ap/(2p+1)) mod (2p+1).
In a letter to Gauss, Germain provides an explanation of her theorem.
First, she states her theorem that, "If the Fermat equation for exponent p has a solution, and if θ (θ=2p+1), is a prime number with no nonzero consecutive pth power residues modulo θ, then θ necessarily divides one of the numbers x, y, or z.
To prove this, the contrary must be assumed, where θ=2p+1 does not divide x, y, or z (mod 2p+1). Also, let a= inverse x (mod 2p+1). We know that a is non-zero because if θ does not divide x, which would have made it 0 mod θ, then the inverse of x is also not divisible by θ and therefore non-zero.
Multiply the original equation by ap
(ax)p+(ay)p≡(az)p (mod 2p+1)
Which reduces to
1+(ay)p=(az)p (mod 2p+1).
This means that the residues, or remainders, of (ay) and (az) will be consecutive and nonzero since θ does not divide a, y, or z. This contradicts the assumption on θ, proving the assertion ( 195,196).
Germain succeeded in proving Fermats Last Theorem for all cases of p less than 100 (Granville and Monagan 329). Germain also had many other works pertaining to FLT. Others followed in her footsteps and proved the theorem for larger cases of p.
People have spent hundreds of years working on this one little proof that was just recently solved. Even Wiles, the author of the proof, said that it had no useful real world application in the Nova documentary, The Proof. Its amazing to see so much work into something so seemingly trivial. On June 27, 1831, at the age of 55, Sophie Germain lost her fight with breast cancer. Her death certificate did not list her as a mathematician or a scientist, but as a rentier, although she was awarded an honorary degree at the University of G_ttengen. Even after her death, she was not recognized as the revolutionary she was. Dauben said in his review of a biography of her life
Germain was possibly the most lackluster of the great or near-great women, not only of her own day but of the recent history of mathematics as well.
He goes on to describe her shortcomings and even suggests changing the title of the book from Sophie Germain: An Essay in the History of the Theory of Elasticity to An Essay in the History of the Theory of Elasticity: the Minor Role of Sophie Germain. Luckily, other authors saw the good role she played in math and science and did not write her off as easily as Dauben, although this may account for the lack of material on her. Even though she was not the best in her field, she attempted more than any of her peers and actually got them interested in what, at the time, was thought to be impossible.
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