Germain and Her Fearless Attempt to Prove Fermat's Last Theorem

TL;DR

This paper explores Sophie Germain's historical efforts and mathematical contributions towards proving Fermat's Last Theorem, highlighting her pioneering work and the significance of her theorem in number theory.

Contribution

It provides a detailed historical account of Germain's work and her partial proofs related to Fermat's Last Theorem, emphasizing her role as a pioneering woman mathematician.

Findings

Germain proved Case 1 of Fermat's Last Theorem for certain prime exponents.

Introduction of Sophie Germain's Theorem relating to prime exponents.

Historical recognition of Germain's contributions to number theory.

Abstract

Two centuries ago, Sophie Germain began to work on her grand plan to prove the theorem of Fermat, the famous conjecture that $x^n + y^n = z^n$ is impossible for nonzero integral values of $x$, $y$, and $z$, when $n > 2$. At that time, this was an open question since nobody knew whether Fermat's assertion was true. Euler had proved it for $n = 3$ and $n = 4$. However, no one else had demonstrated the general case. Then Sophie Germain valiantly entered the world of mathematics in 1804, reaching out to Gauss (writing under the assumed name Monsieur Le Blanc) boldly stating that she could do it. Eventually, Germain conceived a formidable plan for proving Fermat's Last Theorem in its entirety, and in the process she obtained proofs of Case 1 for particular families of exponents. Her efforts resulted in Sophie Germain's Theorem that proves Case 1 of FLT for an odd prime exponent $p$ whenever $2p + 1$ is prime. Today, a prime $p$ is called a Sophie Germain prime if $2p + 1$ is also prime. It remains an unanswered question whether there are an infinite number of Sophie Germain primes. But there is more that Germain did in number theory, much of which was veiled by the mathematicians with whom she shared her work. This article provides historical details of Sophie Germain's efforts, written with the sole intention of paying homage to the only woman mathematician who contributed to proving the most famous assertion of Fermat.