Sophie Germain

:This article is about the mathematician Marie-Sophie Germain. See also Sophie Germain primes.

Name	Marie-Sophie Germain
Caption	Marie-Sophie Germain
Birth date	April 01, 1776
Birth place	Rue Saint-Denis, Paris, France
Death date	June 27, 1831
Other names	Auguste Antoine Le Blanc
Known for	elasticity theory, differential geometry, and number theory
Occupation	mathematician
Nationality	French

Marie-Sophie Germain (April 1, 1776 – June 27, 1831) was a French mathematician, physicist, and philosopher. Despite initial opposition from her parents and difficulties presented by a gender-biased society, she gained education from books in her father's library and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss. One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject. Her work on Fermat's Last Theorem provided a foundation for mathematicians exploring the subject for hundreds of years after. Because of her gender, she was unable to make a career out of mathematics, but worked independently throughout her life.

Early life

Family

Marie-Sophie Germain was born on April 1, 1776, in Paris, France, in a house on Rue Saint-Denis. According to most sources, her father, Ambroise-Franҫois, was a wealthy silk merchant; however, Mary Gray, chair of the Department of Mathematics and Statistics at American University in Washington, D.C., says that he was a goldsmith. In 1789, he was elected as a representative of the bourgeoisie to the États-Généraux, which he saw change into the Constitutional Assembly. It is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy.

Marie-Sophie had one younger sister, named Angélique-Ambroise, and one older sister, named Marie-Madeline. Her mother was also named Marie-Madeline, and this plethora of Maries may have been the reason she went by Sophie. Germain's nephew Armand-Jacques Lherbette, Marie-Madeline's son, published some of Germain's work after she died (see Work in Philosophy). Here she found J. E. Montucla's L'Histoire des Mathématiques, and his story of the death of Archimedes intrigued her. So she pored over every book on mathematics in her father's library, even teaching herself Latin and Greek so she could read works like those of Sir Isaac Newton and Leonhard Euler. She also enjoyed Traité d'Arithmétique by Étienne Bézout and Le Calcul Différential by Jacques Antoine-Joseph Cousin. Later, Cousin visited her in her house, encouraging her in her studies.

Germain's parents did not at all approve of her sudden fascination with mathematics, which was then thought inappropriate for a woman. When night came, they would deny her warm clothes and a fire for her bedroom to try to keep her from studying, but after they left she would take out candles, wrap herself in quilts and do mathematics. As UC Irvine's Women's Studies professor Lynn Osen describes, when her parents found Sophie “asleep at her desk in the morning, the ink frozen in the ink horn and her slate covered with calculations,” they realized that their daughter was serious and relented. After some time, her mother even secretly supported her. As a woman, Germain was barred from attending, but the new system of education made the “lecture notes available to all who asked." Germain obtained the lecture notes and began sending her work to Joseph Louis Lagrange, a faculty member. She used the name M. LeBlanc, When Lagrange saw the intelligence of M. LeBlanc, he requested a meeting, and thus Sophie was forced to disclose her true identity. Fortunately, Lagrange did not mind that Germain was a woman,

Early work in number theory

Correspondence with Legendre

Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres. After studying the work, she opened correspondence with him on number theory, and later, elasticity. Legendre showed some of Germain's work in the Supplément to his second edition of the Théorie des Nombres, where he calls it très ingénieuse (See Best Work on Fermat's Last Theorem).

Correspondence with Gauss

Germain's interest in number theory was renewed when she read Carl Friedrich Gauss' monumental work Disquisitiones Arithmeticae. she wrote, again under the pseudonym of M. LeBlanc, The first letter, dated 21 November 1804, discussed Gauss' Disquisitiones and presented some of Germain's work on Fermat's Last Theorem. In the letter, Germain claimed to have proved the theorem for n = p – 1, where p is a prime number of the form p = 8k + 7; however, her proof contained a weak assumption. Gauss' reply did not comment on Germain's proof.), the French were occupying the German town of Braunschweig, where Gauss lived. Germain, concerned that he might suffer the fate of Archimedes, wrote to General Pernety, a family friend, requesting that he ensure Gauss' safety. He replied,

How can I describe my astonishment and admiration on seeing my esteemed correspondent M leBlanc metamorphosed into this celebrated person. . . when a woman, because of her sex, our customs and prejudices, encounters infinitely more obstacles than men in familiarising herself with [number theory's] 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.

Gauss' letters to Olbers show that his praise for Germain was sincere. In the same 1807 letter, Sophie claimed that if xⁿ + yⁿ is of the form h² + nf², then x + y is also of that form. Gauss replied with a counterexample: 15¹¹ + 8¹¹ can be written as h² + 11f², but 15 + 8 cannot. and in 1809 the letters ceased.

Work in elasticity

Germain's first attempt for the Academy Prize

When Germain's correspondence with Gauss ceased, she took interest in a contest The object of the competition, as stated by the Academy, was “to give the mathematical theory of the vibration of an elastic surface and to compare the theory to experimental evidence." Lagrange's comment that a solution to the problem would require the invention of a new branch of analysis deterred all but two contestants, Denis Poisson and Germain. and leaving Germain as the only entrant to the competition. She submitted her paper early in the fall of 1811, and did not win the prize. The judging commission felt that “the true equations of the movement were not established,” even though “the experiments presented ingenious results.”

Subsequent attempts for the Prize

The contest was extended by two years, and Germain decided to try again for the prize. At first Legendre continued to offer support, but then he refused all help. She did not appear at the ceremony to receive her award. Sophie had derived the correct differential equation, but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, This put the Academy in an awkward position, as they felt the paper to be “inadequate and trivial,” but they did not want to “treat her as a professional colleague, as they would any man, by simply rejecting the work.” In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem. It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence.

In the letter, Germain said that number theory was her preferred field, and that it was in her mind all the time she was studying elasticity. Germain's letter to Gauss contained the first substantial progress toward a proof in 200 years.

Best work on Fermat's Last Theorem

Fermat's Last Theorem is commonly divided into two cases. Case 1 involves all p that do not divide any of x, y, or z. Case 2 includes all p that divide at least one of x, y, or z. Germain proposed the following, commonly called “Sophie Germain's Theorem”:

Let p be an odd prime. If there exists an auxiliary prime P = 2Np + 1 such that: #if x^p + y^p + z^p = 0 (mod P) then P divides xyz, and #p is not a p^th power residue (mod P). Then the first case of Fermat's Last Theorem holds true for p.

Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p<100, but according to Andrea del Centina, “she had actually shown that it holds for every exponent p<197.”

In an unpublished manuscript entitled Remarque sur l’impossibilité de satisfaire en nombres entiers a l’équation x^p + y^p = z^p, around 40 digits long. Sophie did not publish this work. Her brilliant theorem is known only because of the footnote in Legendre's treatise on number theory, where he used it to prove Fermat's Last Theorem for p = 5 (see Correspondence with Legendre).

Two of her philosophical works, Pensées diverses and Considérations générales sur l'état des sciences et des letteres aux différentes epoques de leur culture, were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them. Pensées is a history of science and mathematics with Sophie's commentary.

Final years

In 1829 Germain learned she had breast cancer. Despite the pain, she continued to work. In 1831 Crelle's Journal published her paper on the curvature of elastic surfaces and “a note about finding y and z in $\backslash frac\{4(x^p\; -1)\}\{x-1\}\; =\; y^2\; \backslash pm\; pz^2$." (property holder), not a “mathematicienne."

Honors

Memorials

Germain's resting place in the Père Lachaise Cemetery in Paris is marked by a crumbling gravestone. At the centennial celebration of her life, a street and a girl's school were named after her, and a plaque was placed at the house where she died. The school houses a bust commissioned by the Paris City Council.

A Sophie Germain prime is a prime p such that 2p + 1 is also prime. when k₁ and k₂ are the maximum and minimum values of the normal curvature. H. J. Mozans, whose biography of Germain “is inaccurate and the notes and bibliography are unreliable," but is nonetheless interesting, quotes the mathematician Claude-Louis Navier as saying, “it is a work which few men are able to read and which only one woman was able to write."

Her contemporaries also had good things to say relating to her work in mathematics. Osen relates that “Baron de Prony called her the Hypatia of the nineteenth century,” and “J.J Biot wrote, in the Journal de Savants, that she had probably penetrated the science of mathematics more deeply than any other of her sex." Gauss certainly thought highly of her, and he recognized that European culture presented special difficulties to a woman in mathematics (see Correspondence with Gauss).

Modern critisicms

The modern view generally acknowledges that although Sophie had great talent as a mathematician, her haphazard education had left her without the strong base she needed to truly excel. As explained by Gray, “Germain's work in elasticity suffered generally from an absence of rigor, which might be attributed to her lack of formal training in the rudiments of analysis." Petrovich adds, “This proved to be a major handicap when she could no longer be regarded as a young prodigy to be admired but was judged by her peer mathematicians.”

Notwithstanding the problems with Sophie's theory of vibrations, Gray states that “Germain's work was fundamental in the development of a general theory of elasticity.” Gray adds that “The inclination of sympathetic mathematicians to praise her work rather than to provide substantive criticism from which she might learn was crippling to her mathematical development."

Popular culture

Germain is referenced and quoted in David Auburn's 2001 play Proof. The protagonist is a young struggling female mathematician, who found great inspiration in the work of Germain.

Germain is also mentioned in John Madden's 2005 movie "Proof" in a conversation between Gwyneth Paltrow and Jake Gyllenhaal's characters.

In the fictional work "The Last Theorem" by Arthur C. Clarke and Frederik Pohl, Sophie Germain is credited with inspiring Ranjit Subramanian to solve Fermat's Last Theorem.

See also

Proof of Fermat's Last Theorem for specific exponents

Citations

References

External links

"Sophie Germain", Biographies of Women Mathematicians, Agnes Scott College

Voici ce que j'ai trouvé:": Sophie Germain's grand plan to prove Fermat's Last Theorem 2007

Part 1: An Attack on Fermat, Part 2: A Mathematical Tragedy

Dubreil-Jacotin on Sophie Germain from MacTutor

Category:1776 births Category:1831 deaths Category:19th-century mathematicians Category:French mathematicians Category:Women mathematicians Category:Number theorists Category:Deaths from breast cancer Category:Cancer deaths in France