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If both p and 2p+1 are prime, then
p is a Sophie Germain prime.
The first few such 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 odd Germain 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 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.
Euler and Lagrange
proved the following about Sophie Germain primes:
if p 3
(mod 4) and p > 3, then the
prime 2p+1 divides the Mersenne number
M_{p}.
See Also: CunninghamChain Related pages (outside of this work) References:
 Agoh2000
 Agoh, Takashi, "On Sophie Germain primes," Tatra Mt. Math. Publ., 20 (2000) 6573. Number theory (Liptovský Ján, 1999). MR 1845446
 CFJJK2006
 Csajbók, T., Farkas, G., Járai, A., Járai, Z. and Kasza, J., "Report on the largest known Sophie Germain and twin primes," Ann. Univ. Sci. Budapest. Sect. Comput., 26 (2006) 181183. MR 2388687
 Dubner96
 H. Dubner, "Large Sophie Germain primes," Math. Comp., 65:213 (1996) 393396. MR 96d:11008 (Abstract available)
 FG91
 G. Fee and A. Granville, "The prime factors of Wendt's binomial circulant determinant," Math. Comp., 57:196 (1991) 839848. MR 92f:11183
 JR2007
 Jaroma, John H. and Reddy, Kamaliya N., "Classical and alternative approaches to the Mersenne and Fermat numbers," Amer. Math. Monthly, 114:8 (2007) 677687. MR 2354438
 Peretti1987
 Peretti, A., "The quantity of Sophie Germain primes less than x," Bull. Number Theory Related Topics, 11:13 (1987) 8192. MR 995537
 Yates1987
 Yates, Samuel, Sophie Germain primes. In "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991. pp. 882886, MR 1146271
