"Voici ce que j'ai trouve": Sophie Germain's grand plan to prove Fermat's Last Theorem

TL;DR

This paper reevaluates Sophie Germain's manuscripts, revealing her sophisticated, original approach to Fermat's Last Theorem, including algorithms and methods that predate and differ from later discoveries.

Contribution

It uncovers Germain's comprehensive plan and novel algorithms for Fermat's Last Theorem, highlighting her independent and advanced number theory work.

Findings

Germain developed algorithms similar to later discoveries

She had a detailed, original plan for proving Fermat's Last Theorem

Her work included proofs for specific exponent families

Abstract

A study of Sophie Germain's extensive manuscripts on Fermat's Last Theorem calls for a reassessment of her work in number theory. There is much in these manuscripts beyond the single theorem for Case 1 for which she is known from a published footnote by Legendre. Germain had a fully-fledged, highly developed, sophisticated plan of attack on Fermat's Last Theorem. The supporting algorithms she invented for this plan are based on ideas and results discovered independently only much later by others, and her methods are quite different from any of Legendre's. In addition to her program for proving Fermat's Last Theorem in its entirety, Germain also made major efforts at proofs for particular families of exponents. The isolation Germain worked in, due in substantial part to her difficult position as a woman, was perhaps sufficient that much of this extensive and impressive work may never have been studied and understood by anyone.