How did Sophie Germain contribute to research on Fermat’s last theorem?

Sophie Germain made significant contributions to the study of Fermat’s last theorem, which posits that the equation $x^n+y^n=z^n$ has no non-zero integer solutions for $n>2$.1

In 1819, Germain outlined a strategy for a general solution to the theorem in correspondence with Carl Friedrich Gauss. Her most notable achievement was proving the theorem for a special case: when $x$, $y$, $z$, and $n$ are all relatively prime and $n$ is a prime number smaller than 100.1

Although she did not publish this work herself, her findings were included in a supplement to the second edition of Adrien-Marie Legendre’s Théorie des nombres in 1825. Furthermore, her methodology served as the essential foundation for Legendre’s own proof of the theorem for the case where $n=5$.12