Sophie Germain prime: Map

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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, …... .

The largest known Sophie Germain primes are 648621027630345 × 2253824−1 and 620366307356565 × 2253824−1. They both have 76424 decimal digits and were found in November 2009 by Zoltán Járai, Gabor Farkas, Timea Csajbok, János Kasza and Antal Járai. The previous record was set 6 weeks earlier, 607095 × 2176311−1 with 53081 digits, found by Tom Wu using the program LLR.Before that the record was 48047305725 × 2172403−1 with 51910 digits, found by David Underbakke in January 2007 using the programs TwinGen and LLR. And before that, the record was held by the same team as the November 2009 records, 137211941292195 × 2171960−1 with 51780 digits, found in May 2006.

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 . 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.