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| {{About|the mathematician Marie-Sophie Germain|the number theory (set, or predicate)|Sophie Germain prime}}
| | == 珍しい平日 == |
| {{Infobox scientist
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| | name =Marie-Sophie Germain
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| | image = Germain.jpeg
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| | image_size = 180px
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| | caption = Marie-Sophie Germain
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| | birth_date = {{birth date|mf=yes|1776|4|1|}}
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| | birth_place = Rue Saint-Denis, Paris, France
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| | death_date ={{death date and age|mf=yes|1831|6|27|1776|4|1}}
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| | death_place = Paris, France
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| | residence = France
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| | nationality = French
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| | field = [[Mathematician]], [[physicist]], and [[philosopher]]
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| | work_institution =
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| | alma_mater =
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| | doctoral_advisor =
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| | doctoral_students =
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| | known_for = Elasticity theory, [[differential geometry]], and [[number theory]]; e.g. [[Sophie Germain prime]] numbers
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| | prizes =
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| | footnotes = '''Other name:''' Auguste Antoine Le Blanc
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| }}
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| '''Marie-Sophie Germain''' ({{IPA-fr|maʁi sɔfi ʒɛʁmɛ̃}}; April 1, 1776 – June 27, 1831) was a French [[mathematician]], [[physicist]], and [[philosopher]]. Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library and from correspondence with famous mathematicians such as [[Joseph Louis Lagrange|Lagrange]], [[Adrien-Marie Legendre|Legendre]], and [[Carl Friedrich Gauss|Gauss]]. One of the pioneers of [[Elasticity (physics)|elasticity theory]], she won the grand prize from the [[French Academy of Sciences|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.<ref name="DelCentina_g">[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 373.</ref> Because of prejudice against her gender, she was unable to make a career out of mathematics, but she worked independently throughout her life.<ref>[[#Reference-id16|Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics'']] p. 39.</ref> In recognition of her contribution towards advancement of mathematics, an honorary degree was also conferred upon her by [[University of Göttingen]] six years after her death.<ref>[http://www.famous-mathematicians.com/sophie-germain/]</ref> At the Centenary of her life, a street and a girls’ school were named after her, and the Academy of Sciences established The Grand Prix Sophie Germain in her honour.<ref>[http://www.ams.org/notices/200411/people.pdf Mathematics People]</ref>
| | 何かが高貴な職業として、通常の平日はとても普通の人々の心の中に「医療」部門を見直し、なぜなら彼らの強く、その比類のない重要性 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ腕時計 メンズ] 'セックス'のは非常に困難を見た、それは非常にいくつかの謎と畏怖、そして今日である、最後に彼らの異なる経路ものの残りカウチ場所連続ラッシュアウトから帝国を通じて一般の穴のうち、アリ、、、しかし、彼らは、彼らがすべてであるかのように、これらの高貴な地位精錬「医学」部門は、それがあるそれは街の古風な精錬「医学」協会の中で立っている。<br>すべてのお店の中で<br>今日ガマアルクドスは、以前の過去に比べて暖かいベッドから立ち上がっての多くの人々開かれ、その後、大規模な精製の路上で歩いて、お急ぎの方を見て、戸口に立っていた燃えると畏怖でいっぱい「医学」部門、目、 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html casio 腕時計 g-shock]。<br>総会の [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-8.html カシオ gショック 腕時計] '薬'分裂を洗練<br>ここ数日は、ガマアルクドス年、最も活気のある、熱い時期になります [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-11.html 腕時計 メンズ casio]...<br><br>珍しい平日 |
| | | 相关的主题文章： |
| ==Sophie Germain Prize==
| | <ul> |
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| The Sophie Germain Prize, awarded annually by the Foundation Sophie Germain is conferred by the Academy of Sciences in Paris. Its purpose is to crown a researcher who has conducted research in [[fundamental mathematics]]. This award, in the amount of €8,000 was established in 2003, under the auspices of the [[Institut de France]] and its name pays homage to mathematician Sophie Germain. Previous winners have included
| | <li>[http://www.shuangyinghe.com/plus/feedback.php?aid=3 http://www.shuangyinghe.com/plus/feedback.php?aid=3]</li> |
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| * 2003: [[Claire Voisin]], director of research at the Institute of Mathematics of [[Jussieu University]], [[Denis Diderot]]
| | <li>[http://www.sandhillberries.com/cgi-bin/active/guestbook.cgi http://www.sandhillberries.com/cgi-bin/active/guestbook.cgi]</li> |
| * 2004: [[Henri Berestycki]], research director at the Center of Analysis at the [[EHESS]]
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| * 2005: [[Jean-François Le Gall]], professor at [[Pierre-and-Marie-Curie University|UPMC]], seconded to the Department of Mathematics and applications of the École Normale Supérieure in Paris
| | <li>[http://bbs.zhedong.cc/forum.php?mod=viewthread&tid=750696&fromuid=182517 http://bbs.zhedong.cc/forum.php?mod=viewthread&tid=750696&fromuid=182517]</li> |
| * 2006: [[Michael Harris (mathematician)|Michael Harris]], professor at the Institute of Mathematics of Jussieu University, Denis Diderot
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| * 2007: [[Ngo Bao Chau]], a professor in the Department of Mathematics at the [[University of Paris-Sud in Orsay]]
| | </ul> |
| * 2008: [[Hakan Eliasson]], professor at the Institute of Mathematics of Jussieu University, Denis Diderot
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| * 2009: [[Nessim Sibony]], a professor at the University of Paris-Sud in Orsay
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| * 2010: [[Guy Henniart]], a professor at the University of Paris-Sud in Orsay, Department of Mathematics
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| * 2011: [[Yves Le Jan]], professor at the University of Paris-Sud in Orsay, Department of Mathematics
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| ==Early life==
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| ===Family===
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| 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,<ref name="DelCentina_c">[[#Reference-id17|Del Centina, Andrea. “Letters of Sophie Germain preserved in Florence.”]] sec. 1.</ref><ref name="GrayMary_a">[[#Reference-id6|Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 47.</ref><ref>[[#Reference-id1|Moncrief, J. William. "Germain, Sophie."]] p. 130.</ref> though some believe he was a [[goldsmith]].<ref name="GrayMary">[[#Reference-id16|Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics'']] p. 68.</ref> In 1789, he was elected as a representative of the [[bourgeoisie]] to the [[French States-General|États-Généraux]], which he saw change into the [[French Revolution#National Assembly (1789)|Constitutional Assembly]]. It is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy.<ref name="GrayMary" /> Gray proposes that after his political career, Ambroise-Franҫois became the director of a bank; at least, the family remained well-off enough to support Germain throughout her adult life.<ref>[[#Reference-id16|Gray, Mary W. “Sophie Germain.”]] p. 68.</ref>
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| 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 [[Sophie Germain#Work in philosophy|Work in Philosophy]]).<ref name="GrayMary_a" />
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| ===Introduction to mathematics===
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| When Germain was 13, the [[French Revolution#Storming of the Bastille|Bastille]] fell, and the revolutionary atmosphere of the city forced her to stay inside. For entertainment she turned to her father's library.<ref>[[#Reference-id7|Osen, Lynn. ''Women in Mathematics''.]] pp. 83-84.</ref> Here she found [[Jean-Étienne Montucla|J. E. Montucla's]] ''L'Histoire des Mathématiques'', and his story of the death of [[Archimedes]] intrigued her.<ref name="GrayMary_a" />
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| Germain decided that if geometry, which at that time referred to all of pure mathematics,<ref name="GrayMary_a" /> could hold such fascination for Archimedes, it was a subject worthy of study.<ref>[[#Reference-id10|Ogilvie, Marilyn Bailey. ''Women in Science''.]] p. 90.</ref> So she pored over every book on mathematics in her father's library,<ref>[[#Reference-id7|Osen, Lynn. ''Women in Mathematics''.]] p. 84.</ref> even teaching herself Latin and Greek so she could read works like those of [[Isaac Newton|Sir Isaac Newton]] and [[Leonhard Euler]]. She also enjoyed ''Traité d'Arithmétique'' by [[Étienne Bézout]] and ''Le Calcul Différentiel'' by Jacques Antoine-Joseph Cousin. Later, Cousin visited her in her house, encouraging her in her studies.<ref name="GrayMary_b">[[#Reference-id6|Gray, Mary. ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 48.</ref>
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| 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.<ref>[[#Reference-id6|Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] pp. 47-48.</ref> As [[UC Irvine|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.<ref>[[#Reference-id7|Osen, Lynn. ''Women in Mathematics'']] p. 85.</ref> After some time, her mother even secretly supported her.<ref name="GrayMary_b" />
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| ===l'École Polytechnique===
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| [[Image:Porton antigua sede École Polytechnique Sainte Geneviève.JPG|thumb|right|Entrance to the historic building of the École Polytechnique]]
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| In 1794, when Germain was 18, the [[École Polytechnique]] opened.<ref name="Moncrief">[[#Reference-id1|Moncrief, J. William. "Germain, Sophie."]] p. 130</ref> As a woman, Germain was barred from attending, but the new system of education made the “lecture notes available to all who asked."<ref name="GrayMary_b" /> The new method also required the students to “submit written observations."<ref>[[#Reference-id16|Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics'']] p. 69.</ref> Germain obtained the lecture notes and began sending her work to [[Joseph Louis Lagrange]], a faculty member. She used the name of a former student Monsieur Antoine-August Le Blanc,<ref name="GrayMary_b" /><ref>{{cite web|url=http://www.pbs.org/wgbh/nova/physics/sophie-germain.html|author=Simon Singh|year=1997|publisher=WGBH Educational Foundation|title=Math's Hidden Woman}}</ref> “fearing,” as she later explained to Gauss, “the ridicule attached to a female scientist."<ref>[[#Reference-id19|Mackinnon, Nick. “Sophie Germain, or, was Gauss a feminist?”]] p. 348.</ref> 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,<ref name="GrayMary_b" /> and he became her mentor.<ref name="Moncrief" /> He too visited her in her home, giving her moral support.<ref>[[#Reference-id7|Osen, Lynn. ''Woman in Mathematics''.]] p. 85.</ref>
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| ==Early work in number theory==
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| ===Correspondence with Legendre===
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| Germain first became interested in number theory in 1798 when [[Adrien-Marie Legendre]] published ''Essai sur la théorie des nombres''.<ref name="DelCentina">[[#Reference-id17|Del Centina, Andrea. “Letters of Sophie Germain preserved in Florence.”]] sec. 2</ref> 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'' ["very ingenious"] (See [[Sophie Germain#Best work on Fermat's Last Theorem|Best Work on Fermat's Last Theorem]]).<ref>[[#Reference-id8|Sampson, J. H. “Sophie Germain and the Theory of Numbers.”]] p. 158</ref>
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| ===Correspondence with Gauss===
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| [[Image:Carl Friedrich Gauss.jpg|thumb|right|Carl Friedrich Gauss]]
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| Germain's interest in number theory was renewed when she read [[Carl Friedrich Gauss|Carl Friedrich Gauss']] monumental work ''[[Disquisitiones Arithmeticae]]''.<ref name="DelCentina" /> After three years of working through the exercises and trying her own proofs for some of the theorems,<ref>[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 352.</ref> she wrote, again under the pseudonym of M. LeBlanc,<ref name="GrayMary_b" /> to the author himself, who was one year younger than she.<ref>[[#Reference-id8|Sampson, J. H. “Sophie Germain and the Theory of Numbers.”]] p. 157.</ref> The first letter, dated 21 November 1804,<ref>[[#Reference-id16|Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics'']] p. 71.</ref> 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'' = 8''k'' + 7;<ref name="Dickson">[[#Reference-id14|Dickson, L. E. ''History of the Theory of Numbers''.]] p. 733.</ref> however, her proof contained a weak assumption.<ref name="DelCentina_a">[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 355.</ref> Gauss' reply did not comment on Germain's proof.<ref name="DelCentina_a" />
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| Around 1807 (sources differ)<ref name="OsenLynn">[[#Reference-id7|Osen, Lynn. ''Women in Mathematics''.]] p. 88.</ref><ref name="Dunnington">[[#Reference-id13|Dunnington, G. Waldo. ''Carl Gauss: Titan of Science''.]] p. 67.</ref> 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.<ref name="GrayMary_b" /> General Pernety sent a chief of a battalion to meet with Gauss personally to see that he was safe.<ref name="Dunnington" /> As it turned out, Gauss was fine,<ref name="OsenLynn" /> but he was confused by the mention of Sophie's name.<ref name="Dunnington" />
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| Three months after the incident, Germain disclosed her true identity to Gauss.<ref>[[#Reference-id19|Mackinnon, Nick. "Sophie Germain, or, was Gauss a feminist?"]] p. 348.</ref> He replied,
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| <blockquote>
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| 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.<ref>[[#Reference-id19|qtd. in Mackinnon, Nick. “Sophie Germain, or, was Gauss a feminist?”]] p. 349.</ref>
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| </blockquote>
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| Gauss' letters to [[Heinrich Wilhelm Matthäus Olbers|Olbers]] show that his praise for Germain was sincere.<ref name="Dunnington" /><ref>[[#Reference-id11|Bell, E. T. ''Men of Mathematics''.]] p. 262.</ref> In the same 1807 letter, Sophie claimed that if ''x''<sup>''n''</sup> + ''y''<sup>''n''</sup> is of the form ''h''<sup>2</sup> + ''nf''<sup>2</sup>, then ''x'' + ''y'' is also of that form. Gauss replied with a counterexample: 15<sup>11</sup> + 8<sup>11</sup> can be written as ''h''<sup>2</sup> + 11''f''<sup>2</sup>, but 15 + 8 cannot.<ref name="Dickson" />
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| Although Gauss thought well of Germain, his replies to her letters were often delayed, and he generally did not review her work.<ref name="DelCentina_a" /> Eventually his interests turned away from number theory,<ref name="McGillSara">[[#Reference-id2|McGill, Sara Ann. “Sophie Germain.”]] p. 9.</ref> and in 1809 the letters ceased.<ref name="DelCentina_a" /> Despite the friendship of Germain and Gauss, they never met.<ref name="GrayMary_e">[[#Reference-id6|Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 49.</ref>
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| ==Work in elasticity==
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| ===Germain's first attempt for the Academy Prize===
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| [[Image:Echladni.jpg|thumb|left|Ernst Florens Friedrich Chladni]]
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| When Germain's correspondence with Gauss ceased, she took interest in a contest<ref name="McGillSara" /> sponsored by the Paris Academy of Sciences concerning [[Ernst Chladni|Ernst Chladni's]] experiments with vibrating metal plates.<ref>[[#Reference-id9|Ullmann, D. “Life and work of E.F.F. Chladni.”]] p. 27.</ref> 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."<ref name="GrayMary_c">[[#Reference-id6|Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 52.</ref> Lagrange's comment that a solution to the problem would require the invention of a new branch of [[mathematical analysis|analysis]] deterred all but two contestants, [[Siméon Denis Poisson|Denis Poisson]] and Germain.<ref name="GrayMary_c" /> Then Poisson was elected to the Academy, thus becoming a judge instead of a contestant,<ref name="PetrovichVesna">[[#Reference-id4|Petrovich, Vesna Crnjanski. “Women and the Paris Academy of Sciences.”]] p. 384.</ref> and leaving Germain as the only entrant to the competition.<ref name="GrayMary_c" />
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| In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research.<ref name="PetrovichVesna_a">[[#Reference-id4|Petrovich, Vesna Crnjanski. “Women and the Paris Academy of Sciences.”]] p. 386.</ref> 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.”<ref name="PetrovichVesna" /> Lagrange was able to use Germain's work to derive an equation that was “correct under special assumptions.”<ref name="GrayMary_d">[[#Reference-id16|Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics''.]] p. 71.</ref>
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| ===Subsequent attempts for the Prize===
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| 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.<ref name="PetrovichVesna" /> Germain's anonymous<ref name="GrayMary_d" /> 1813 submission was still littered with mathematical errors, especially involving [[multiple integral|double integrals]],<ref name="GrayMary_c" /> and it received only an honorable mention because “the fundamental base of the theory [of elastic surfaces] was not established."<ref name="PetrovichVesna" /> The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson.<ref name="GrayMary_d" /> In 1814 he published his own work on elasticity, and did not acknowledge Germain's help (although he had worked with her on the subject and, as a judge on the Academy commission, had had access to her work).<ref name="GrayMary_c" />
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| Germain submitted her third paper, “Recherches sur la théorie des surfaces élastiques”<ref name="GrayMary_d" /> under her own name, and on 8 January 1816<ref name="GrayMary_c" /> she became the first woman to win a prize from the Paris Academy of Sciences.<ref name="PetrovichVesna_b">[[#Reference-id4|Petrovich, Vesna Crnjanski. “Women and the Paris Academy of Sciences.”]] p. 385.</ref> She did not appear at the ceremony to receive her award.<ref name="GrayMary_d" /> Although Germain had at last been awarded the ''prix extraordinaire'',<ref name="DelCentina_a" /> the Academy was still not fully satisfied.<ref>[[#Reference-id10|Ogilvie, Marilyn Bailey. ''Women in Science''.]] p. 91.</ref> Sophie had derived the correct [[differential equation]],<ref name="Ullmann">[[#Reference-id9|Ullmann, D. “Life and work of E.F.F. Chladni.”]] p. 31.</ref> but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler,<ref name="GrayMary_d" /> which led to incorrect boundary conditions.<ref name="Ullmann" /> Here is Germain's final equation:
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| :<math>N^2\left(\frac{\partial^4 z}{\partial x^4} + \frac{\partial^4 z}{\partial x^2 \partial y^2} + \frac{\partial^4 z}{\partial y^4}\right) + \frac{\partial^2 z}{\partial t^2} = 0 </math>
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| where ''N''<sup>2</sup> is a constant.<ref name="GrayMary_d" />
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| After winning the Academy contest, she was still not able to attend its sessions because of the Academy's tradition of excluding women other than the wives of members. Seven years later this tradition was broken when she made friends with [[Joseph Fourier]], a secretary of the Academy, who obtained tickets to the sessions for her.<ref name="PetrovichVesna_a" />
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| ===Later work in elasticity===
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| Germain published her prize-winning essay at her own expense in 1821, mostly because she wanted to present her work in opposition to that of Poisson. In the essay she pointed out some of the errors in her method.<ref name="GrayMary_d" />
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| In 1826 she submitted a revised version of her 1821 essay to the Academy. According to Andrea del Centina, a math professor at the [[University of Ferrara]] in Italy, the revision included attempts to clarify her work by “introducing certain simplifying hypotheses."<ref name="DelCentina_b">[[#Reference-id17|Del Centina, Andrea. “Letters of Sophie Germain preserved in Florence.”]] sec. 4.</ref> 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.”<ref name="DelCentina_b" /> So [[Augustin-Louis Cauchy]], who had been appointed to review her work, recommended she publish it, and she followed his advice.<ref name="DelCentina_b" />
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| One further work of Germain's on elasticity was published posthumously in 1831: her “Mémoire sur la courbure des surfaces.” She used the [[mean curvature]] in her research (see [[Sophien|Honors in Number Theory]]).<ref name="GrayMary_d" />
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| ==Later work in number theory==
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| ===Renewed interest===
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| Germain's best work was in number theory,<ref name="DelCentina_c" /> and her most significant contribution to number theory dealt with Fermat's Last Theorem.<ref>[[#Reference-id8|Sampson, J. H. “Sophie Germain and the Theory of Numbers.”]] p. 158.</ref> In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem.<ref name="DelCentina_d">[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 357.</ref> It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence.<ref>[[#Reference-id17|Del Centina, Andrea. “Letters of Sophie Germain preserved in Florence.”]] sec. 2.</ref>
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| 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.<ref name="DelCentina_d" /> She outlined a strategy for a general proof of Fermat's Last Theorem, including a proof for a special case (see [[Sophie Germain#Best work on Fermat's Last Theorem|Best Work on Fermat's Last Theorem]]).<ref>[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] pp. 356-57.</ref> Germain's letter to Gauss contained the first substantial progress toward a proof in 200 years.<ref name="McGillSara" /> She asked Gauss if her approach to the theorem was worth pursuing. Gauss never answered.<ref>[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 362.</ref>
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| ===Best work on Fermat's Last Theorem===
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| [[Image:Pierre de Fermat.jpg|thumb|right|[[Pierre de Fermat]]]]
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| 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|Sophie Germain's Theorem]]”:<ref name="DelCentina_f">[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 349.</ref>
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| <blockquote>
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| Let ''p'' be an odd prime. If there exists an auxiliary prime ''P'' = 2''Np'' + 1 such that:
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| #if ''x''<sup>''p''</sup> + ''y''<sup>''p''</sup> + ''z''<sup>''p''</sup> = 0 ([[modular arithmetic|mod ''P'']]) then ''P'' divides ''xyz'', and
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| #''p'' is not a ''p''<sup>th</sup> power residue (mod ''P'').
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| Then the first case of [[Fermat's Last Theorem]] holds true for ''p''.<ref name="DelCentina_e">[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 372.</ref>
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| </blockquote>
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| 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.”<ref name="DelCentina_e" /> L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.<ref>[[#Reference-id14|Dickson, L. E. ''History of the Theory of Numbers''.]] p. 763.</ref>
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| In an unpublished manuscript entitled ''Remarque sur l’impossibilité de satisfaire en nombres entiers a l’équation x<sup>p</sup> + y<sup>p</sup> = z<sup>p</sup>'',<ref name="DelCentina_f" /> Germain showed that any counterexamples to Fermat's theorem for ''p''>5 must be numbers “whose size frightens the imagination,”<ref name="Atdin">[[#Reference-id5|atd. in Cipra, Barry. “A Woman Who Counted.”]] p. 899.</ref> around 40 digits long.<ref>[[#Reference-id18|Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.”]] p. 371.</ref> 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 [[Sophie Germain#Correspondence with Legendre|Correspondence with Legendre]]).<ref name="Atdin" /> Germain also proved or nearly proved several results that were attributed to Lagrange or were rediscovered years later.<ref name="DelCentina_g" /> Del Centina states that “after almost two hundred years her ideas were still central”,<ref name="DelCentina_g" /> but ultimately her method did not work.<ref name="Atdin" />
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| ==Work in philosophy==
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| In addition to mathematics, Germain studied philosophy and [[psychology]].<ref name="GrayMary_b" /> She wanted to classify facts and generalize them into laws that could form a system of psychology and [[sociology]], which were then just coming into existence. Her philosophy was highly praised by [[Auguste Comte]].<ref>[[#Reference-id16|Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics''.]] p. 73.</ref>
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| Two of her philosophical works, ''Pensées diverses'' and ''Considérations générales sur l'état des sciences et des lettres, aux différentes époques de leur culture'',<ref name="OsenLynn_a">[[#Reference-id7|Osen, Lynn. ''Women in Mathematics''.]] p. 91.</ref> were published, both posthumously. This was due in part to the efforts of Lherbette, her nephew, who collected her philosophical writings and published them.<ref>[[#Reference-id6|Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 53.</ref> ''Pensées'' is a history of [[science]] and mathematics with Sophie's commentary.<ref name="OsenLynn_a" /> In ''Considérations'', the work admired by Comte, Sophie argues that there are no differences between the sciences and the [[humanities]].<ref name="OgilvieMarilyn">[[#Reference-id10|Ogilvie, Marilyn Bailey. ''Women in Science''.]] p. 92.</ref>
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| ==Final years==
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| In 1829 Germain learned she had [[breast cancer]]. Despite the pain,<ref>[[#Reference-id17|Del Centina, Andrea. “Letters of Sophie Germain preserved in Florence.”]] sec. 5-6.</ref> she continued to work. In 1831 ''[[Crelle's Journal]]'' published her paper on the [[curvature]] of elastic surfaces and “a note about finding {{math|''y''}} and {{math|''z''}} in <math>\textstyle \frac{4(x^p -1)}{x-1} = y^2 \pm pz^2</math>."<ref name="GrayMary_d" /> And American University's Gray records, “She also published in ''Annales de chimie et de physique'' an examination of principles which led to the discovery of the laws of equilibrium and movement of elastic solids."<ref name="GrayMary_d" /> On June 27 of 1831, she died in the house at 13 rue de Savoie.<ref name="GrayMary_e" />
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| Despite Germain's intellectual achievements, her death certificate lists her as a “rentière – annuitant”<ref name="Mozans">[[#Reference-id15|Mozans, H. J. ''Woman in Science''.]] p. 156.</ref> (property holder<ref name="GrayMary_g">[[#Reference-id6|Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 50.</ref>), not a “mathematicienne."<ref name="Mozans" /> But her work was not unappreciated by everyone. When the matter of honorary degrees came up at the [[University of Göttingen]] six years after Germain's death, Gauss lamented, “[Germain] proved to the world that even a woman can accomplish something worthwhile in the most rigorous and abstract of the sciences and for that reason would well have deserved an honorary degree."<ref>[[#Reference-id19|qtd. in Mackinnon, Nick. “Sophie Germain, or, was Gauss a feminist?”]] p. 347.</ref>
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| ==Honors==
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| ===Memorials===
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| [[File:Grave, Sophie Germain.jpg|thumb|right|Grave of Sophie Germain in Père Lachaise Cemetery]]
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| Germain's resting place in the [[Père Lachaise Cemetery]] in Paris is marked by a crumbling gravestone.<ref name="GrayMary_e" /><ref name="GrayMary_f">[[#Reference-id16|Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics''.]] p. 68.</ref><!-- The grave seems "OK" in the image, and the lower plaque includes a reference to the girls' school named after her, so I assume it has been kept up or replaced since the centennial mentioned next. --> At the centennial celebration of her life, a street and a girls' 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.<ref name="GrayMary_f" />
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| ===Honors in number theory===
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| E. Dubouis defined a ''sophien'' of a prime {{math|''n''}} to be a prime {{math|θ}} where {{math|1=θ = ''kn'' + 1}}, for such {{math|''n''}} that yield {{math|θ}} such that {{math|1=''x''<sup>''n''</sup> = ''y''<sup>''n''</sup> + 1 (mod θ)}} has no solutions when {{math|''x''}} and {{math|''y''}} are prime to {{math|''n''}}.<ref>[[#Reference-id14|Dickson, L. E. ''History of the Theory of Numbers''.]] p. 769.</ref>
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| A [[Sophie Germain prime]] is a prime {{math|''p''}} such that {{math|2''p'' + 1}} is also prime.<ref name="DelCentina_e" />
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| The ''Germain curvature'' (also called [[mean curvature]]) is <math>\frac{k_1 + k_2}{2}</math>,<ref>[[#Reference-id19|Mackinnon, Nick. “Sophie Germain, or, was Gauss a feminist?”]] p. 347.</ref> when {{math|''k''<sub>1</sub>}} and {{math|''k''<sub>2</sub>}} are the maximum and minimum values of the normal curvature.<ref name="GrayMary_d" />
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| ''Sophie Germain's Identity'' states that for any {{math|{''x,y''}}}, then,
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| :<math>x^4+4y^4 = ((x+y)^2+y^2)((x-y)^2+y^2) = (x^2+2xy+2y^2)(x^2-2xy+2y^2). \,</math>
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| ==Criticisms==
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| === Contemporary praise and criticisms ===
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| Vesna Petrovich, a graduate of the [[University of Michigan]], found that the educated world's response to the publication in 1821 of Germain's prize-winning essay “ranged from polite to indifferent".<ref name="PetrovichVesna_b" /> Yet, some critics had high praise for it. Of her essay in 1821, Cauchy said, “[it] was a work for which the name of its author and the importance of the subject both deserved the attention of mathematicians."<ref>[[#Reference-id6|qtd. in Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 49.</ref> Germain was also included in [[H. J. Mozans|H. J. Mozans']] book "[[Woman in Science]]",<ref>[[#Reference-id15|qtd. in Mozans, H. J. ''Woman in Science''.]]</ref> although Marilyn Bailey Ogilvie claims that the biography "is inaccurate and the notes and bibliography are unreliable".<ref>[[#Reference-id10|Ogilvie, Marilyn Bailey. ''Women in Science''.]] p. 201.</ref> Nevertheless, it 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."<ref>[[#Reference-id15|qtd. in Mozans, H. J. ''Woman in Science''.]] p. 156.</ref>
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| Germain's contemporaries also had good things to say relating to her work in mathematics. Osen relates that “[[Gaspard de Prony|Baron de Prony]] called her the [[Hypatia of Alexandria|Hypatia]] of the nineteenth century,” and “J.J Biot wrote, in the [[Journal des sçavans|''Journal de Savants'']], that she had probably penetrated the science of mathematics more deeply than any other of her sex."<ref>[[#Reference-id7|Osen, Lynn. ''Women in Mathematics''.]] p. 90.</ref> Gauss certainly thought highly of her, and he recognized that European culture presented special difficulties to a woman in mathematics (see [[Sophie Germain#Correspondence with Gauss|Correspondence with Gauss]]).
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| ===Modern praise and criticisms===
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| The modern view generally acknowledges that although Germain 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."<ref>[[#Reference-id6|Gray, Mary. “Sophie Germain.” ''Women of Mathematics: A Bibliographic Sourcebook''.]] p. 51.</ref> Petrovich adds, “This proved to be a major handicap when she could no longer be regarded as a young [[child prodigy|prodigy]] to be admired but was judged by her peer mathematicians.”<ref>[[#Reference-id4|Petrovich, Vesna Crnjanski. “Women and the Paris Academy of Sciences.”]] pp. 384-85.</ref>
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| Not withstanding the problems with Germain's theory of vibrations, Gray states that “Germain's work was fundamental in the development of a general theory of elasticity.”<ref name="GrayMary_c" /> [[H. J. Mozans]] writes, however, that when the [[Eiffel tower]] was built and the architects inscribed the names of 72 great French scientists, Germain's name was not among them: despite the salience of her work to the towers construction “Was she excluded from this list... because she was a woman? It would seem so."<ref name="Mozans" />
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| Concerning her early work in number theory, J. H. Sampson, author of “Sophie Germain and the Theory of Numbers,” states, “She was clever with formal algebraic manipulations; but there is little evidence that she really understood the ''Disquisitiones'', and her work of that period that has come down to us seems to touch only on rather superficial matters."<ref>[[#Reference id8|Sampson, J. H. “Sophie Germain and the Theory of Numbers.”]] p. 158.</ref> 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."<ref name="GrayMary_g" /> Yet Marilyn Bailey Ogilvie, Curator of the History of Science Collections and Professor of the History of Science at the [[University of Oklahoma]] recognizes that “Sophie Germain's creativity manifested itself in pure and applied mathematics...[she] provided imaginative and provocative solutions to several important problems,"<ref name="OgilvieMarilyn" /> and, as Petrovich proposes, it may have been her very lack of training that gave her unique insights and approaches.<ref name="PetrovichVesna_b" /> Louis Bucciarelli and Nancy Dworsky, Germain's biographers, summarize as follows: “All the evidence argues that Sophie Germain had a mathematical brilliance that never reached fruition due to a lack of rigorous training available only to men."<ref>[[#Reference-id4|qtd. in Petrovich, Vesna Crnjanski. “Women and the Paris Academy of Sciences.”]] p. 386.</ref>
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| ==Germain in popular culture==
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| Germain was referenced and quoted in David Auburn's 2001 play ''[[Proof (play)|Proof]].'' The protagonist is a young struggling female mathematician, Catherine, who found great inspiration in the work of Germain. Germain was also mentioned in John Madden's film adaptation of the same play "[[Proof (2005 film)]]" in a conversation between Catherine (Gwyneth Paltrow) and Hal (Jake Gyllenhaal).
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| In the fictional work "[[The Last Theorem]]" by [[Arthur C. Clarke]] and [[Frederik Pohl]], Sophie Germain was credited with inspiring Ranjit Subramanian to solve [[Fermat's Last Theorem]].
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| ==See also==
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| *[[Proof of Fermat's Last Theorem for specific exponents]]
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| ==Citations==
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| {{reflist|2}}
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| ==References==
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| {{Library resources box|by=yes|onlinebooksby=yes|viaf=12440398}}
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| *{{wikicite|id=id11|reference=Bell, E. T. ''Men of Mathematics''. New York: Simon and Schuster, 1965. Print.}}
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| *{{wikicite|id=id5|reference=Cipra, Barry. “A Woman Who Counted.” ''Science'' 319.5865 (2008): 899. Web. Sept. 2009.}}
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| *{{wikicite|id=id17|reference=Del Centina, Andrea. “Letters of Sophie Germain preserved in Florence.” ''Historia Mathematica'' 32.1 (2005): 60-75. Web. Sept. 2009.}}
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| *{{wikicite|id=id18|reference=Del Centina, Andrea. “Unpublished manuscripts of Sophie Germain and a revaluation of her work on Fermat's Last Theorem.” ''Archive for History of Exact Sciences'' 62.4 (2008): 349-392. Web. Sept. 2009.}}
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| *{{wikicite|id=id14|reference=[[L. E. Dickson|Dickson, L. E.]] ''[[History of the Theory of Numbers]]''. Vol. 2, 732-769. New York: G. E. Stechert, 1934. Print. 2 vols.}}
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| *{{wikicite|id=id13|reference=Dunnington, G. Waldo. ''Carl Gauss: Titan of Science''. New York: Hafner, 1955. Print.}}
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| *{{wikicite|id=id16|reference=Gray, Mary W. “Sophie Germain.” ''Complexities: Women in Mathematics''. Ed. Bettye Anne Case and Anne M. Leggett. United Kingdom: Princeton University Press, 2005. 68-75. Print.}}
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| *{{cite encyclopedia |ref=Reference-id6 |last=Gray |first=Mary |title=Sophie Germain (1776-1831) |encyclopedia=Women of Mathematics: A Bibliographic Sourcebook |editors=Louise S. Grinstein and Paul Campbell |publisher=Greenwood |year=1978 |isbn=978-0-313-24849-8 |pages=47–55}}
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| *{{wikicite|id=id19|reference=Mackinnon, Nick. “Sophie Germain, or, was Gauss a feminist?” ''The Mathematical Gazette'' 74.469 (1990): 346-351. Web. Sept. 2009.}}
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| *{{wikicite|id=id2|reference=McGill, Sara Ann. “Sophie Germain.” ''History Remembers Scientists of the Past'' (2000): 9. Web. Sept. 2009.}}
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| *{{wikicite|id=id1|reference=Moncrief, J. William. "Germain, Sophie." ''Mathematics''. Ed. Barry Max Brandenberger, Jr.. Vol. 2. New York: Macmillan Reference USA, 2002. 103. Web. 15 Sept. 2009 4 vols.}}
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| *{{cite book |ref=Reference-id15 |last=Mozen |first=H. J. (pseud.)|authorlink=John Augustine Zahm |title=Women in Science: With an Introductory Chapter on Women's Long Struggle for Things of the Mind|url=http://www.archive.org/details/womaninscience00zahmrich |publisher=D. Appleton |year=1913 |pages=154–157}}
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| *{{wikicite|id=id10|reference=Ogilvie, Marilyn Bailey. ''Women in Science''. Cambridge: MIT Press, 1986. Print.}}
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| *{{wikicite|id=id7|reference=Osen, Lynn. ''Women in Mathematics''. Cambridge: MIT Press, 1984. Print.}}
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| *{{wikicite|id=id4|reference=Petrovich, Vesna Crnjanski. “Women and the Paris Academy of Sciences.” ''Eighteenth-Century Studies'' 32.3 (1999): 383-391. Web. Sept. 2009.}}
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| *{{wikicite|id=id8|reference=Sampson, J. H. “Sophie Germain and the Theory of Numbers.” ''Archive for History of Exact Sciences'' 41.2 (1990): 157-161. Print.}}
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| *{{wikicite|id=id9|reference=Ullmann, D. “Life and work of E.F.F. Chladni.” ''European Physical Journal – Special Topics'' 145.1 (2007): 25-32. Web. Sept. 2009.}}
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| *{{wikicite|id=id3|reference=[[William C. Waterhouse|Waterhouse, William C.]] “A counterexample for Germain.” ''American Mathematical Monthly'' 101.2 (1994): 140. Web. Sept. 2009.}}
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| ==External links==
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| *[http://www.agnesscott.edu/lriddle/women/germain.htm "Sophie Germain", Biographies of Women Mathematicians], [[Agnes Scott College]]
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| *{{MacTutor Biography|id=Germain}}
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| *{{MathGenealogy|id=55175}}
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| *[http://www.math.nmsu.edu/%7Edavidp/germain.pdf Voici ce que j'ai trouvé:": Sophie Germain's grand plan to prove Fermat's Last Theorem] 2007
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| *[http://www.sciencenews.org/articles/20080223/mathtrek.asp Part 1: An Attack on Fermat], [http://www.sciencenews.org/articles/20080301/mathtrek.asp Part 2: A Mathematical Tragedy]
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| *[http://www-groups.dcs.st-and.ac.uk/~history/Extras/Dubreil-Jacotin_Germain.html Dubreil-Jacotin on Sophie Germain] from MacTutor
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| {{Authority control|VIAF=12440398}}
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| {{Persondata <!-- Metadata: see [[Wikipedia:Persondata]]. -->
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| | NAME = Germain, Sophie
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| | ALTERNATIVE NAMES =
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| | SHORT DESCRIPTION = Mathematician
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| | DATE OF BIRTH = April 1, 1776
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| | PLACE OF BIRTH = Paris, France
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| | DATE OF DEATH = June 27, 1831
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| | PLACE OF DEATH = Paris, France
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| }}
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| {{DEFAULTSORT:Germain, Sophie}}
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| [[Category:1776 births]]
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| [[Category:1831 deaths]]
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| [[Category:19th-century French mathematicians]]
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| [[Category:Women mathematicians]]
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| [[Category:Number theorists]]
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| [[Category:Deaths from breast cancer]]
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| [[Category:Cancer deaths in France]]
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| [[Category:Physicists]]
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| [[Category:Women physicists]]
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