A prime p is said to be a Sophie Germain prime if both p and 
are prime. The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ... (Sloane's A005384). The numbers of Sophie Germain primes less than 
for n = 1, 2, ... are 3, 10, 37, 190, 1171, 7746, 56032, ... (Sloane's A092816).
The largest known Sophie Germain prime is 
,
Sophie Germain primes p of the form 
(which makes a prime) correspond to the indices of composite Mersenne numbers 
.
Around 1825, Sophie Germain proved that the first case of Fermat's last theorem is true for such primes, i.e., if p is a Sophie Germain prime, then there do not exist integers x, y, and z different from 0 and none a multiple of p such that
Cunningham Chain, Fermat's Last Theorem, Mersenne Number, Twin Primes
Caldwell, C. K. "Prime Pages. The Top Twenty: Sophie Germain." http://primes.utm.edu/top20/page.php?id=2.
Dubner, H. "Large Sophie Germain Primes." Math. Comput. 65, 393-396, 1996.
Hoffman, P. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, p. 190, 1998.
Indlekofer, K. H. and Járai, A. "Largest Known Twin Primes and Sophie Germain Primes." Math. Comput. 68, 1317-1324, 1999.
Ribenboim, P. "Sophie Germain Primes." §5.2 in The New Book of Prime Number Records. New York: Springer-Verlag, pp. 329-332, 1996.
Shanks, D. Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, pp. 154-157, 1993.
Sloane, N. J. A. Sequences A005384/M0731 and A092816 in "The On-Line Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/.