Sophie Germain, math\'ematicienne extraordinaire: A story stranger than fiction

TL;DR

Sophie Germain was a pioneering woman mathematician whose recent rediscovered work significantly advanced number theory and Fermat's Last Theorem, revealing her overlooked contributions and innovative methods.

Contribution

The paper uncovers Sophie Germain's previously unknown extensive research on Fermat's Last Theorem, highlighting her innovative approaches and long-term plan for proof.

Findings

Germain's manuscripts reveal a comprehensive plan to prove Fermat's Last Theorem.

Her work involved new tools like congruence and modular primitive roots.

Recent discoveries show her contributions were more substantial than previously recognized.

Abstract

Sophie Germain (1776-1831) was the first woman we know who did important original research in mathematics, specifically in elasticity theory and number theory. Celebrating her semiquincentennial year, we outline Germain's recently unearthed number theory results on Fermat's Last Theorem, in the context of her life, work, and interactions with Lagrange, Legendre, and Gauss. For two centuries her accomplishment on Fermat's Last Theorem was thought to consist of a single theorem attributed to her in a publication by Legendre, the first general result towards proving Fermat's Last Theorem. But recent discoveries in her handwritten manuscripts and correspondence with Legendre and Gauss show that she accomplished much more, albeit forgotten. In particular, she had a grand plan for proving Fermat's Last Theorem in its entirety, and carried this plan a long way, using then new tools, e.g., congruence, modular primitive roots, and permutations.