Marie-Sophie Germain

1. Germain's Identity

A factorization identity.

$$ a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab) $$

2. Sophie Germain Primes

A prime \(p\) where \(2p+1\) is also prime, crucial in her work on Fermat's Last Theorem.

$$ p \in \mathbb{P}, \quad 2p+1 \in \mathbb{P} $$