Sophie Germain's theorem

Statement. If $p$ is a Sophie Germain prime (i.e. $2p+1$ is prime) and there exist integers $x,y,z$ with $x^p+y^p=z^p$ and $p\nmid xyz$, then $2p+1$ divides $xyz$. In particular, no solution with $p\nmid xyz$ exists.

Application. This proves the first case of Fermat's Last Theorem for exponents $p$ that are Sophie Germain primes (e.g. $p=3,5,11,23,29,41,\ldots$).

Conclusion. Sophie Germain's theorem was a major early contribution to FLT.