Safe and Sophie Germain primes
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 × 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.
The first few Sophie Germain primes are:
- 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, …... (sequence A005384 in the OEIS).
The largest known Sophie Germain prime is 48047305725 × 2172403−1. It has 51910 decimal digits and was found by David Underbakke on January 25, 2007 using the programs TwinGen and LLR. [1] The previous record was 137211941292195 × 2171960−1, which has 51780 decimal digits and was found by Járai et al. on May 3, 2006. [2]
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. 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 1 or more Sophie Germain primes, ending with a prime which does not have to be a Sophie Germain, 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 2p−1.
Sophie Germain primes were mentioned in the stage play Proof and the subsequent film.
Application in (pseudo-)random number generation
Sophie Germain primes have a practical application in the generation of pseudo-random numbers. The decimal expansion of 1/q will produce a stream of q−1 pseudo-random digits, if q is the safe prime of a Sophie Germain prime p, with p congruent to 3, 9, or 11 (mod 20). Thus “suitable” prime numbers q are 7, 23, 47, 59, 167, 179, etc (corresponding to p = 3, 11, 23, 29, 83, 89, etc.). The result is a stream of length q−1 digits (including leading zeros); for more see OEIS sequence OEIS: A000355. So, for example, using q = 23 generates the pseudo-random digits 0, 4, 3, 4, 7, 8, 2, 6, 0, 8, 6, 9, 5, 6, 5, 2, 1, 7, 3, 9, 1, 3. Note that these digits are not appropriate for cryptographic purposes, as the value of each can be derived from its predecessor in the digit-stream.
References
- Maximally Periodic Reciprocals, R.A.J. Matthews (1992). Bulletin of the Institute of Mathematics and its Applications; vol 28 pp 147-148.
External links
- The Top Twenty Sophie Germain Primes — from the Prime Pages.