Encyclopedia > Sophie Germain prime

  Article Content

Sophie Germain prime

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.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Northampton, Suffolk County, New York

... For every 100 females age 18 and over, there are 82.9 males. The median income for a household in the town is $47,500, and the median income for a family is $35,893. ...

 
 
 
This page was created in 30.8 ms