Sophie Germain prime (another Prime Pages' Glossary entries)
 Glossary: Prime Pages: Top 5000: The hardware and software on this system was updated September 4th.  Please let me know of any problem you encounter. 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 Mp. See Also: CunninghamChainRelated pages (outside of this work) Sophie Germain primes (records and theorems) Euler and Lagrange's theorem (proof of the theorem mentioned above) Math's Hidden Woman (about the woman Sophie Germain)References: Agoh2000 Agoh, Takashi, "On Sophie Germain primes," Tatra Mt. Math. Publ., 20 (2000) 65--73.  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) 181--183.  MR 2388687 Dubner96 H. Dubner, "Large Sophie Germain primes," Math. Comp., 65:213 (1996) 393--396.  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) 839--848.  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) 677--687.  MR 2354438 Peretti1987 Peretti, A., "The quantity of Sophie Germain primes less than x," Bull. Number Theory Related Topics, 11:1-3 (1987) 81--92.  MR 995537 Yates1987 Yates, Samuel, Sophie Germain primes.  In "The mathematical heritage of C. F. Gauss," World Sci. Publ., River Edge, NJ, 1991.  pp. 882--886, MR 1146271 Chris K. Caldwell © 1999-2014 (all rights reserved)