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." L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.

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-François, was a wealthy silk merchant, though some believe he was a goldsmith. In 1789, he was elected as a representative of the bourgeoisie to the États-Généraux, which he saw change into the Constitutional Assembly. It is therefore assumed that Sophie witnessed many discussions between her father and his friends on politics and philosophy. Gray proposes that after his political career, Ambroise-François became the Director of a bank; at least, the family remained well-off enough to support Germain throughout her adult life.

In 1794, when Germain was 18, the École Polytechnique opened. As a woman, Germain was barred from attending, but the new system of education made the "lecture notes available to all who asked." The new method also required the students to "submit written observations." 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, "fearing," as she later explained to Gauss, "the ridicule attached to a female scientist." 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, and he became her mentor. He too visited her in her home, giving her moral support.

Germain first became interested in number theory in 1798 when Adrien-Marie Legendre published Essai sur la théorie des nombres. 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 Her work on Fermat's Last Theorem).

Germain's interest in number theory was renewed when she read Carl Friedrich Gauss' monumental work Disquisitiones Arithmeticae. After three years of working through the exercises and trying her own proofs for some of the theorems, she wrote, again under the pseudonym of M. LeBlanc, to the author himself, who was one year younger than she. The first letter, dated 21 November 1804, 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 = 8k + 7. However, her proof contained a weak assumption, and Gauss' reply did not comment on Germain's proof.

Gauss' letters to Olbers show that his praise for Germain was sincere. In the same 1807 letter, Sophie claimed that if x + y is of the form h + nf, then x + y is also of that form. Gauss replied with a counterexample: 15 + 8 can be written as h + 11f, but 15 + 8 cannot.

In 1809 Germain began work. Legendre assisted by giving her equations, references, and current research. 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." Lagrange was able to use Germain's work to derive an equation that was "correct under special assumptions."

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. Germain's anonymous 1813 submission was still littered with mathematical errors, especially involving double integrals, and it received only an honorable mention because "the fundamental base of the theory [of elastic surfaces] was not established." The contest was extended once more, and Germain began work on her third attempt. This time she consulted with Poisson. 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).

Germain's best work was in number theory, and her most significant contribution to number theory dealt with Fermat's Last Theorem. In 1815, after the elasticity contest, the Academy offered a prize for a proof of Fermat's Last Theorem. It reawakened Germain's interest in number theory, and she wrote to Gauss again after ten years of no correspondence.

Germain submitted her third paper, "Recherches sur la théorie des surfaces élastiques" under her own name, and on 8 January 1816 she became the first woman to win a prize from the Paris Academy of Sciences. She did not appear at the ceremony to receive her award. Although Germain had at last been awarded the prix extraordinaire, the Academy was still not fully satisfied. Sophie had derived the correct differential equation, but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions. Here is Germain's final equation:

Vesna Petrovich found that the educated world's response to the publication in 1821 of Germain's prize-winning essay "ranged from polite to indifferent". 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." Germain was also included in H. J. Mozans' book "Woman in Science", although Marilyn Bailey Ogilvie claims that the biography "is inaccurate and the notes and bibliography are unreliable". 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."

In 1826 she submitted a revised version of her 1821 essay to the Academy. According to Andrea Del Centina, the revision included attempts to clarify her work by "introducing certain simplifying hypotheses." 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." So Augustin-Louis Cauchy, who had been appointed to review her work, recommended she publish it, and she followed his advice.

In 1829 Germain learned she had breast cancer. Despite the pain, she continued to work. In 1831 Crelle's Journal published her paper on the curvature of elastic surfaces and "a note about finding y and z in ${\displaystyle \textstyle {\frac {4(x^{p}-1)}{x-1}}=y^{2}\pm pz^{2}}$." Mary 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." On 27 June 1831, she died in the house at 13 rue de Savoie.

The Germain curvature (also called mean curvature) is ${\displaystyle {\frac {k_{1}+k_{2}}{2}}}$, when k₁ and k₂ are the maximum and minimum values of the normal curvature.

Germain was referenced and quoted in David Auburn's 2001 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 in a conversation between Catherine (Gwyneth Paltrow) and Hal (Jake Gyllenhaal).

The Sophie Germain Prize (Prix Sophie Germain), awarded annually by the Foundation Sophie Germain, is conferred by the Academy of Sciences in Paris. Its purpose is to honour a French Mathematician for research in the foundations of mathematics. This award, in the amount of €8,000, was established in 2003, under the auspices of the Institut de France.

Despite Germain's intellectual achievements, her death certificate lists her as a "rentière – annuitant" (property holder), not a "mathématicienne." But her work was not unappreciated by everyone. When the matter of honorary degrees came up at the University of Göttingen in 1837—six years after Germain's death—Gauss lamented, "she [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."