How did Sophie Germain contribute to work on Fermat's last theorem?

Sophie Germain made significant strides toward a general solution for 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, she outlined a strategy to address the theorem and successfully proved a special case: when $x$, $y$, $z$, and $n$ are all relatively prime and $n$ is a prime number smaller than 100. 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. Her methodology was instrumental in Legendre’s subsequent proof of the theorem for the case where $n=5$.12

While Germain’s work provided a vital foundation for 19th-century progress, the theorem remained unresolved until it was finally proven for all cases by Andrew Wiles in 1995.1