If both
p and 2
p+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 method's 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 C
2 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 Mp.
(Thanks to Chip Kerchner for the last two entries in the table above.)
Definitions and Notes