High Quality Content by WIKIPEDIA articles! In number theory, a prime number p is 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 x 23 + 1 = 47, also prime. These numbers are named after French mathematician Marie-Sophie Germain. A Sophie Germain prime p > 3 is of the form 6k?1 or, equivalently, p ? 5 (mod 6) — as is its matching safe prime 2p+1. We note that the other form for a prime p > 3 is 6k+1 or, equivalently, p ? 1 (mod 6), and that 3|(2p+1) — thus excluding such p from the Sophie Germain prime domain. This is trivially proven using modular arithmetic. It is conjectured that there are infinitely many Sophie Germain primes, but like the twin prime conjecture, this has not been proven.

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