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Sophie Germain prime

Redirected from Germain Primes A prime p is 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 an interval [1, 10⁴] (SIDN A005384 (http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?Anum=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 C₂ x / (log x)² where C₂ is the twin prime constant, approximately 0.660161. For x=10,000 an estimation gives us approximately 413 Sophie Germain primes, which is still too inaccurate.

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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