# Sophie Germain prime

A prime number p is called a Sophie Germain prime if 2p + 1 is also prime. For example, 23 is a Sophie Germain prime because it is a prime and 2 × 23 + 1 = 47, also prime.

These primes acquired significance because of Sophie Germain's proof that Fermat's last theorem is true for such primes. It is conjectured that there are infinitely many Sophie Germain primes, but like the twin prime conjecture, this has not been proven. The first few Sophie Germain primes are (sequence A005384 in OEIS):

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, ...

Currently, the largest known Sophie Germain prime is 7068555 × 2121301 - 1, discovered by Predrag Minovic in January 2005, using TwinGen and LLR.

A heuristic estimate for the number of Sophie Germain primes less than n is 2C2 n / (ln n)2 where C2 is the twin prime constant, approximately 0.660161. For n   =   104, this estimate predicts 156 Sophie Germain primes, which has a 20% error compared to the exact value of 190 above. For n = 107, the estimate predicts 50822, which is still 10% off from the exact value of 56032.

A sequence {p, 2p + 1, 2(2p + 1) + 1, ...} of Sophie Germain primes is called a Cunningham chain of the first kind. Every term of such a sequence except the first and last is both a Sophie Germain prime and a safe prime.

It was the eponymous proof in the stage play Proof.