While Sophie Germain has been best known today for her work on elasticity, she made important contributions to number theory as well.
Sophie Germain's claim to number theoretic fame rests on a footnote in Legendre's Théorie des Nombres, crediting her with what is known today as Sophie Germain's Theorem, the first general result toward a proof of Fermat's Last Theorem. She used it to prove Case I of Fermat's Last Theorem for all prime exponents less than 100, and today her methods have been generalized to apply to an infinite number of exponents. Germain, however, never published her theorem, describing it instead in correspondence with Legendre and Gauss; it was detailed by Legendre when he published his own solution for exponent five. It has been generally assumed that she was the junior partner in a collaboration with Legendre.
However, a reevaluation of her manuscripts, and her correspondence
Legendre and Gauss ("Voice ce que j'ai trouvé:"), indicates
Not only did she develop the general version of her theorem
but she also deserves credit for vast additional work on Fermat's Last
Theorem, much of it previously
attributed to Legendre. See our preprint "Voici
ce que j'ai trouve'': Sophie Germain's Grand Plan to Prove Fermat's
Last Theorem (to appear in Historia Mathematica)
for all the newly discovered details of Germain's work
on Fermat's Last Theorem. Some
of her original writings were also presented in translation in our book
Expeditions: Chronicles by the Explorers. We would like to find
more of the Legendre-Germain correspondence, and would welcome any
of its existence.