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A prime number p is called a Sophie Germain prime
if 2p + 1 is also prime. They acquired significance because of Sophie Germain's proof that Fermat's
last theorem is true for such primes. 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 OEIS).
A heuristic estimate 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 above. For n = 107, the estimate predicts 50822, whichs is still 10% off from
the exact value of 56032.
A sequence {p, 2p + 1, 2(2p + 1) + 1, ...} of Sophie Germain primes
is called a Cunningham chain of the first kind.
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