What are they?

A Sophie Germain prime is a prime such that $2p+1$ is also prime - for example, 23 is a Sophie Germain prime since 47 is also prime.

The largest known Sophie Germain prime has close to 400,000 digits; it is conjectured that there are infinitely many such primes, although this has not yet been proved.

Relatedly, a Cunningham chain is a sequence of numbers such that $u_0$ is a Sophie Germain prime, and $u_{n+1} = 2u_n + 1$ if $u_n$ is a prime - for example, ${ 2, 5, 11, 23, 47, 95}$ is a Cunningham chain. Another unproved conjecture is that there are Cunningham chains of any length you like (although we know there aren’t infinitely long ones.)

Why are they important?

Germain used primes of this form in her work on Fermat’s Last Theorem - she was the first to attack FLT with a grand plan rather than a piecemeal approach. Her particular idea embodies “Wrong, but useful”: it didn’t work, but it led to interesting things.

In modern times, Sophie Germain primes (and their related ‘safe primes’) are important cryptographically: the products of unsafe primes are vulnerable to various factorisation methods such as Pollard’s rho - and similar problems exist when looking at cryptographic systems based on the discrete logarithm problem.

Sophie Germain primes can also be used as a basic random number generator: the decimal expansion of $\frac{1}{q}$ produces a stream of $q-1$ pseudo-random digits if $q$ is the safe prime of a Sophie Germain prime $p$, such that $p$ is congruent to 3, 9 or 11 (modulo 20). (This is one of my favourite ways to generate ‘random’ numbers without a calculator.)

Who was Germain?

Marie-Sophie Germain was born in Paris in 1776. Denied a career as a mathematician ((because the universities of the time decreed that women had no business doing manly maths)), she worked independently, corresponding with Gauss, Lagrange and Legendre, working under the pseudonym of Auguste Le Blanc until Lagrange requested a meeting with the young ‘man’ of unusual intelligence.

She died, also in Paris, in 1831.