Barred from the Paris lecture halls, Sophie Germain studied under a man’s name and corresponded with Gauss as ‘Monsieur LeBlanc.’ She found a whole class of primes now bearing her name: a prime p where 2p+1 is ALSO prime — the key to a big early dent in Fermat’s Last Theorem, and today the backbone of safe cryptographic primes. Slide and hunt them.
A Sophie Germain prime is a prime p for which 2p+1 (the ‘safe prime’) is also prime: 2, 3, 5, 11, 23, 29, 41, 53, 83, 89… Germain used them to prove Fermat’s Last Theorem for a whole family of exponents when almost no general progress existed. The instrument tests both primality live. A fail-loud self-check throws unless 23 and 89 register as Germain primes and 7 does not (2·7+1=15 is composite) — the exact double-primality condition.
Whether infinitely many Sophie Germain primes exist is an OPEN conjecture (like the twin primes). The primality tests and the definition are exact; the crypto use (safe primes resist certain factoring attacks) is standard.