Sophie Germain prime

In number theory, a prime number p is a Sophie Germain prime if 2p + 1 is also prime. The number 2p + 1 associated with a Sophie Germain prime is called a safe prime. For example, 29 is a Sophie Germain prime and 2 × 29 + 1 = 59 is its associated safe prime. Sophie Germain primes are named after French mathematician Sophie Germain, who used them in her investigations of Fermat's Last Theorem. Sophie Germain primes and safe primes have applications in public key cryptography and primality testing. It has been conjectured that there are infinitely many Sophie Germain primes, but this remains unproven.

The first few Sophie Germain primes are: (less than 1000)

In cryptography much larger Sophie Germain primes like ${\displaystyle 1,846,389,521,368+11^{600}}$ are required.

Two distributed computing projects, PrimeGrid and Twin Prime Search, include searches for large Sophie Germain primes.

The largest known Sophie Germain primes as of March 2016^[update] are:

It is conjectured that there are infinitely many Sophie Germain primes, but this has not been proven. Several other famous conjectures in number theory generalize this and the twin prime conjecture; they include the Dickson's conjecture, Schinzel's hypothesis H, and the Bateman–Horn conjecture.

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