What was Sophia Germain's claim to fame?

Answer 1

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 with Legendre and Gauss ("Voice ce que j'ai trouvé:"), indicates otherwise. Not only did she develop the general version of her theorem independently, but she also deserves credit for vast additional work on Fermat's Last Theorem, much of it previously attributed to Legendre. See our submitted preprint "Voici ce que j'ai trouve'': Sophie Germain's Grand Plan to Prove Fermat's Last Theorem 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 Mathematical Expeditions: Chronicles by the Explorers. We would like to find more of the Legendre-Germain correspondence, and would welcome any knowledge of its existence.