Sophie Germain prime
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. There are 190 Sophie Germain primes in the interval [1, 104] (SIDN A005384):
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131,
173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443,
491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911,
953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451,
1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973,
2003, 2039, 2063, 2069, 2129, 2141, 2273, 2339, 2351, 2393, 2399, 2459,
2543, 2549, 2693, 2699, 2741, 2753, 2819, 2903, 2939, 2963, 2969, 3023,
3299, 3329, 3359, 3389, 3413, 3449, 3491, 3539, 3593, 3623, 3761, 3779,
3803, 3821, 3851, 3863, 3911, 4019, 4073, 4211, 4271, 4349, 4373, 4391,
4409, 4481, 4733, 4793, 4871, 4919, 4943, 5003, 5039, 5051, 5081, 5171,
5231, 5279, 5303, 5333, 5399, 5441, 5501, 5639, 5711, 5741, 5849, 5903,
6053, 6101, 6113, 6131, 6173, 6263, 6269, 6323, 6329, 6449, 6491, 6521,
6551, 6563, 6581, 6761, 6899, 6983, 7043, 7079, 7103, 7121, 7151, 7193,
7211, 7349, 7433, 7541, 7643, 7649, 7691, 7823, 7841, 7883, 7901, 8069,
8093, 8111, 8243, 8273, 8513, 8663, 8693, 8741, 8951, 8969, 9029, 9059,
9221, 9293, 9371, 9419, 9473, 9479, 9539, 9629, 9689, 9791
A heuristic estimate for the number of Sophie Germain primes less than x is 2C2 x / (ln x)2 where C2 is the twin prime constant, approximately 0.660161. For x=104, this estimate predicts 156 Sophie Germain primes, which has a 20% error compared to the exact value of 190 above. (For x=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.
Referenced By
Complete Cunningham chain | Cunningham chain | Cunningham chain of the first kind | Cunningham chain of the second kind | Eight hundred | Eighty-nine | Eighty-three | Fermat | Fermat's Last Theorem | Fermat conjecture | FermatsLastTheorem | Fermats Last Theorem | Fifty-three | Forty-one | Generalized Cunningham chain | List of mathematical topics (S-U) | List of number theory topics | List of prime numbers | Nine hundred | One hundred thirteen | PierreDeFermat | Pierre de Fermat | Prime number | Prime numbers | Primes | Rader's FFT algorithm | Twenty-nine | Twenty-three | Wiles's theorem
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