Germain was born to a middle-class merchant family in Paris, France, and began studying mathematics at age thirteen, despite her parents' strong attempts to dissuade her from engaging in a 'men's profession'. Several years later, she managed to get some lecture notes from several courses at École Polytechnique, a school which did not admit women.
Germain was particularly interested in Joseph-Louis 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 and became her mentor.
In 1804 she began corresponding with Carl Friedrich Gauss, again using her pseudonym, after reading his famous Disquisitiones Arithmeticae (1801). He eventually learned her true identity in 1806, when Napoleon Bonaparte was invading Prussia and Gauss's birthplace, Brunswick. Fearful that Gauss would meet a fate like that of Archimedes, Germain requested that General Pernety, a friend of hers, personally ensure Gauss's safety. The general explained to Gauss that Germain had asked that he be protected, which confused Gauss since he had never heard of her. She then wrote to him admitting she was female, to which he responded:
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.
In 1811 Germain entered the French Academy of Sciences's contest to explain the underlying mathematical law of a German mathematician, attempting to explain Ernst 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. She became the first female to attend sessions at the French Academy of Sciences—excepting the wives of other members.
One of Germain's major contributions to number theory was the following theorem: if x, y, and z are integers, and x5 + y5 = z5 then either x, y, or z has to be divisible by five. This proof, which she first described in a letter to Gauss, became quite significant as it restricted the possible solutions of Fermat's last theorem. One significant contribution is the concept of the Sophie Germain prime, which is a prime number p where 2p+1 is also prime. She proved that Fermat's last theorem is true for such primes.
- O'Connor, John J., and Edmund F. Robertson. "Sophie Germain". MacTutor History of Mathematics archive.
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