Despite initial opposition from her parents and difficulties presented by society, she gained education from books in her father's library, including ones by Euler, and from correspondence with famous mathematicians such as Lagrange, Legendre, and Gauss (under the pseudonym of Monsieur Le Blanc).

One of the pioneers of elasticity theory, she won the grand prize from the Paris Academy of Sciences for her essay on the subject.

Gray proposes that after his political career, Ambroise-François became the director of a bank; in any case, the family remained well-off enough to support Germain throughout her adult life.

[5] Germain thought that if the geometry method, which at that time referred to all of pure mathematics,[5] could hold such fascination for Archimedes, it was a subject worthy of study.

[8] 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 Différentiel by Jacques Antoine-Joseph Cousin.

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.

[11] 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-Auguste Le Blanc,[9][12] "fearing", as she later explained to Gauss, "the ridicule attached to a female scientist".

[6] Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres.

Legendre included 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").

[15] Germain's interest in number theory was renewed when she read Carl Friedrich Gauss's monumental work Disquisitiones Arithmeticae.

[13] He replied:[22] 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.

[25] When Germain's correspondence with Gauss ceased, she took interest in a contest sponsored by the Paris Academy of Sciences concerning Ernst Chladni's experiments with vibrating metal plates.

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.

[26] Germain's anonymous[18] 1813 submission was still littered with mathematical errors, especially involving double integrals,[27] and it received only an honorable mention because "the fundamental base of the theory [of elastic surfaces] was not established".

[27] Germain submitted her third paper, "Recherches sur la théorie des surfaces élastiques",[18] under her own name, and on 8 January 1816[27] she became the first woman to win a prize from the Paris Academy of Sciences.

Seven years later this situation was transformed, when she made friends with Joseph Fourier, a secretary of the academy, who obtained tickets to the sessions for her.

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".

[40] In an unpublished manuscript titled Remarque sur l'impossibilité de satisfaire en nombres entiers a l'équation xp + yp = zp,[38] Germain showed that any counterexamples to Fermat's theorem for p > 5 must be numbers "whose size frightens the imagination",[41] around 40 digits long.

[44] Pensées is a collection of personal notes on scientific subjects (the writings of Tycho, Newton, and Laplace), aphorisms, and philosophical reflections.

[7] In January 2020, Satellogic, a high-resolution Earth observation imaging and analytics company, launched a ÑuSat type micro-satellite named in honor of Sophie Germain.

[25] Germain was also included in H. J. Mozans' 1913 book Woman in Science,[54] although Marilyn Bailey Ogilvie claims that the biography "is inaccurate and the notes and bibliography are unreliable".

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.

"[56] 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.

"[15] Gray adds on to say "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.

"[50] Yet Marilyn Bailey Ogilvie recognizes that "Sophie Germain's creativity manifested itself in pure and applied mathematics ... [she] provided imaginative and provocative solutions to several important problems",[47] and, as Petrovich proposes, it may have been her very lack of training that gave her unique insights and approaches.

Germain was also mentioned in John Madden's film adaptation of the same name in a conversation between Catherine (Gwyneth Paltrow) and Hal (Jake Gyllenhaal).

A library book entitled Sophie Germain: The Unsolved Riddle (there appears to be no such book) serves as a "mailbox" for a brilliant, deceased prime number theorist named Safiya Zamil to have passed a handwritten note to a future mathematician on a similar prime number quest.