Editing Sophie Germain (section)

==Early work in number theory==

===Correspondence with Legendre===
Germain first became interested in [[number theory]] in 1798 when [[Adrien-Marie Legendre]] published {{lang|fr|Essai sur la théorie des nombres}}.{{sfn|Del Centina|2005|loc=sec. 2}} After studying the work, she opened correspondence with him on number theory, and later, [[elasticity (physics)|elasticity]]. Legendre included some of Germain's work in the {{lang|fr|Supplément}} to his second edition of the {{lang|fr|Théorie des Nombres}}, where he calls it {{lang|fr|très ingénieuse}} ("very ingenious"). See also [[#Her work on Fermat's Last Theorem|Her work on Fermat's Last Theorem]] below.{{sfn|Sampson|1990|p=158}}

===Correspondence with Gauss===
Germain's interest in number theory was renewed when she read [[Carl Friedrich Gauss]]'s monumental work {{lang|la|[[Disquisitiones Arithmeticae]]}}.{{sfn|Del Centina|2005|loc=sec. 2}} After three years of working through the exercises and trying her own proofs for some of the theorems,{{sfn|Del Centina|2008|page=352}} she wrote, again under the pseudonym of M. Le Blanc,{{sfn|Gray|1978|p=48}} to the author himself, who was one year younger than she.{{sfn|Sampson|1990|p=157}} Gauss's replies were mailed to the home of [[Antoine Isaac Silvestre de Sacy|Antoine-Isaac, Baron Silvestre De Sacy]], who must have understood Germain's reasons for assuming a masculine pseudonym and agreed to help her conceal her identity.{{sfn|Musielak|2020|p=201}}

The first letter, dated 21 November 1804,{{sfn|Gray|2005|p=71}} discussed Gauss's {{lang|la|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''&nbsp;=&nbsp;''p''&nbsp;−&nbsp;1, where ''p'' is a [[prime number]] of the form ''p''&nbsp;=&nbsp;8''k''&nbsp;+&nbsp;7.{{sfn|Dickson|1919|p=733}} However, her proof contained a weak assumption, and Gauss's reply did not comment on Germain's proof.{{sfn|Del Centina|2008|page=355}}

Around 1807 (sources differ),{{sfn|Dunnington|1955|p=67}} during the Napoleonic wars, 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 ([[Joseph Marie de Pernety]]), a family friend, requesting that he ensure Gauss's safety. {{sfn|Gray|1978|p=48}} General Pernety sent the chief of a battalion to meet with Gauss personally to see that he was safe.{{sfn|Dunnington|1955|p=67}} As it turned out, Gauss was fine, but he was confused by the mention of Sophie's name.{{sfn|Dunnington|1955|p=67}}

Three months after the incident, Germain disclosed her true identity to Gauss.{{sfn|Mackinnon|1990|p=348}}  He replied:{{sfn|Mackinnon|1990|p=349}}
{{blockquote|
How can I describe my astonishment and admiration on seeing my esteemed correspondent M. Le Blanc 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 noblest courage, extraordinary talent, and superior genius.
}}[[Image:Carl Friedrich Gauss.jpg|thumb|right|1840 painting of Carl Friedrich Gauss; by Gottlieb Biermann after [[Christian Albrecht Jensen]]]]Gauss's letters to [[Heinrich Wilhelm Matthäus Olbers|Olbers]] show that his praise for Germain was sincere.{{sfn|Dunnington|1955|p=67}}{{sfn|Bell|1937|p=262}} In the same 1807 letter, Germain claimed that if <math>x^n + y^n</math> is of the form <math>h^2 + nf^2</math>, then <math>x + y</math> is also of that form. Gauss replied with a counterexample: <math>15^{11} + 8^{11}</math> can be written as <math>h^2 + 11 f^2</math>, but <math>15 + 8</math> cannot.{{sfn|Dickson|1919|p=733}}{{sfn|Waterhouse|1994}}

Although Gauss thought well of Germain, his replies to her letters were often delayed, and he generally did not review her work.{{sfn|Del Centina|2008|page=355}} Eventually his interests turned away from number theory, and in 1809 the letters ceased.{{sfn|Del Centina|2008|page=355}} Despite the friendship of Germain and Gauss, they never met.{{sfn|Gray|1978|p=49}}