Sophie Germain was born to a middle-class merchant family on April 1, 1776 in Paris, France and began studying mathematics at age 13, despite strong attempts to dissuade her from engaging in a 'men's profession' by her parents. Several years later, she managed to get some lecture notes from several courses at Ecole Polytechnique which she studied despite the school not admitting females.

Germain was particularly interest in Joseph Lagrange's teachings and submitted papers and assignments under the pseudonym Monsieur Le Blanc, a former student of Lagrange's. Lagrange was so impressed by the paper that he asked to meet Le Blanc and Germain was forced to reveal her identity to him. Lagrange apparently considered her a talented mathematician, not just a dumb female and became her mentor.

In 1804 she began corresponding with the almighty Karl Friedrich Gauss, again using her pseudonym, after reading his Disquisitiones Arithmeticae. He eventually learned her true identity after she requested that General Pernety, a friend of hers, ensure his safety when Napoleon Bonaparte was invading Prussia. The General explained to Gauss that Germain had asked that he be protected, which freaked Gauss since he'd never heard of this Germain person. She then wrote to him admitting she was female, which he thought was quite cool.

But how to describe to you my admiration and astonishment at seeing my esteemed correspondent Monsieur Le Blanc metamorphose himself into this illustrious personage who gives such a brilliant example of what I would find it difficult to believe. A taste for the abstract sciences in general and above all the mysteries of numbers is excessively rare: one is not astonished at it: the enchanting charms of this sublime science reveal only to those who have the courage to go deeply into it. But when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and superior genius. Indeed nothing could prove to me in so flattering and less equivocal manner that the attractions of this science, which has enriched my life with so many joys, are not chimerical, the predilection with which you have honored it.

One of Germain's major contributions to number theory--which she described to Gauss in a letter--was a mathematical proof which eventually proved valuable in proving Fermat's Last Theorem: if x, y, and z are integers, and x5 + y5 = z5 then either x, y, or z has to be divisible by five, thus restricting possible solutions to Fermat's Last Theorem.

However, in 1808 Gauss was appointed professor of astronomy at the University of Göttingen and his interest shifted to applied mathematics and he stopped replying to her letters.

In 1811 Germain entered The French Academy of Sciences' contest to explain the underlying mathematical law of a German mathematician, attempting to explain Ernst Florens Friedrich Chladni's study on vibrations of elastic surfaces. After failing twice she finally won in 1816, thus bringing her into the ranks of great mathematicians and she became the first female to attend sessions at the French Academy of Sciences, excepting the wives of other members.

Her central contributions to mathematics were in the fields of number theory and elasticity theory. One significant item is the concept of Germain Primes. (A prime p where 2p+1 is also prime.)

With prompting from Gauss, in 1830 the University of Göttingen agreed to award Germain with an honourary degree but before she received it she died of breast cancer on June 27, 1831.

Sources
Fermat's Enigma by Simon Singh. (Great book.)
http://www.agnesscott.edu/lriddle/women/germain.htm
http://www.pbs.org/wgbh/nova/proof/germain.html
http://www.mathnews.uwaterloo.ca/BestOf/WomenInMath6905.html

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