The Top Twenty: Sophie Germain (p)

The Prime Pages keeps a list of the 5000 largest known primes, plus a few each of certain selected archivable forms and classes. These forms are defined in this collection's home page. This page is about one of those forms. Comments and suggestions requested.

If both p and 2p+1 are prime, then p is a Sophie Germain prime. The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, and 131. Around 1825 Sophie Germain proved that the first case of Fermat's Last Theorem is true for such primes. Soon after Legendre began to generalize this by showing the first case of FLT also holds for odd primes p such that kp+1 is prime, k=4, 8, 10, 14 and 16. In 1991 Fee and Granville [FG91] extended this to k<100, k not a multiple of three. Many similar results were also shown, but now that Fermat's Last Theorem has been proven by Wiles, they are of less interest.

Are there infinitely many Sophie Germain primes? Ribenboim indicates that the sieve methods of Brun (see the twin primes page) can be used to estimate that the number of primes p < x for which kp+a is prime is bounded above by C x/(log x)² (so they have density zero among the primes). Heuristically, it seems reasonable to conjecture that there is a lower bound of this form as well. More specifically (see a simple heuristic), it is conjectured that the number of Sophie Germain primes less than N is asympototic to

where C₂ is the twin prime constant (estimated by Wrench and others to be approximately 0.6601618158...). This estimate works suprisingly well! For example:

The number of Sophie Germain
primes less than N
N	actual	estimate
1,000	37	39
100,000	1171	1166
10,000,000	56032	56128
100,000,000	423140	423295
1,000,000,000	3308859	3307888
10,000,000,000	26569515	26568824

Euler and Lagrange proved that if we also have p 3 (mod 4) and p > 3, then 2p+1 is prime (and p is a Sophie Germain prime) if and only if 2p+1 divides the Mersenne M_p.