Sophie Germain Prime Search This is the next addition to PrimeGrid's ever expanding prime search projects. The Sophie Germain Prime search honors Marie-Sophie Germain, an extraordinary French mathematician who made important contributions to the fields of differential geometry and number theory, and to the study of Fermat's Last Theorem. A prime number p is called a Sophie Germain prime if 2p + 1 is also prime. For example, 5 is a Sophie Germain prime because it is prime and 2 × 5 + 1 = 11, is also prime.
We'll be searching the form k*2^n-1. If it is prime, then we'll check k*2^n+1, k*2^(n-1)-1, & k*2^(n+1)-1. We are able to do this because a quad sieve was performed for this search. This sieve ensured that k*2^n-1, k*2^n+1, k*2^(n-1)-1, & k*2^(n+1)-1 did not have any small prime divisors. The opportunity to find SG's and Twins in the same sieve file is appealing. However, we "expect" to find a Sophie Germain prime first.
This quad sieve was prepared quite some time ago; so it was readily available. Here are some stats for the search: (incomplete)
k range: k>41T n=1290000 sieve depth: ??? candidates remaining: 34,190,344
Probability of one or more significant pair = 80.1% Probability of one or more SG = 66.7% Probability of one or more Twin = 42.3%
Approximate WU length: Ryzen 5 2nd gen 2.0GHz - ~1200 secs (~20min) per core Core i5 Skylake 2.0GHz - ~900 secs (~15min) per core C2Q 3.2 GHz - ~1200 secs (~20.0 minutes) per 4 cores
For more information about Sophie Germain primes, please visit these links: http://primes.utm.edu/glossary/page.php?sort=SophieGermainPrime http://mathworld.wolfram.com/SophieGermainPrime.html http://en.wikipedia.org/wiki/Sophie_Germain_prime