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1/2p

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 numbers are named after French mathematician Marie-Sophie Germain. more...

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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. 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 137211941292195 × 2¹⁷¹⁹⁶⁰ - 1, which has 51780 decimal digits and was found by Járai et al. on May 3, 2006 (Underwood 2006; 121301 - 1, discovered by Predrag Minovic in January 2005, using TwinGen and LLR.

A heuristic estimate (due to G. H. Hardy and J. E. Littlewood) 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.

If a Sophie Germain prime p is congruent to 3 mod 4, then its matching safe prime 2p + 1 will be a divisor of the Mersenne number 2^p - 1.

Sophie Germain primes were the subject of the eponymous proof in the stage play Proof and the subsequent film Proof.

Application in random number generation

Sophie Germain primes have a practical application in the generation of random numbers. The decimal expansion of reciprocal 1/q will produce a stream of pseudorandom numbers of length q - 1 if q is the safe prime of a Sophie Germain prime S, with S congruent to 3, 9, or 11 mod 20. Thus “suitable” prime numbers q are 7, 23, 47, 59, 167, 179, etc (corresponding to S = 3, 11, 23, 29, 83, 89, etc.). The result is a stream of length q-1 digits (including leading zeros). So, for example, using q = 23 generates the random digits 0,4,3,4,7,8,2,6,0,8,6,9,5,6,5,2,1,7,3,9,1,3