A prime is said to be a Sophie Germain prime if both 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). It is not known if there are an infinite number of Sophie Germain primes (Hoffman 1998, p. 190).

The numbers of Sophie Germain primes less than for , 2, ... are 3, 10, 37, 190, 1171, 7746, 56032, ... (Sloane's A092816).

The largest known Sophie Germain prime as of Feb. 2009 has , which has 51910 decimal digits and was found by Chris Caldwell on Jan. 25, 2007 (http://primes.utm.edu/primes/page.php?id=79261).

The definition of Sophie Germain primes and the value of the largest (then-)known such prime were mentioned by the characters Hal and Catherine in the 2005 film Proof.

Sophie Germain primes of the form 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 is a Sophie Germain prime, then there do not exist integers , , and different from 0 and none a multiple of such that

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 Erdős 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.

Minovic, P. "RE: Record Sophie Germain Prime Pair." Jan 11, 2005. http://groups.yahoo.com/group/primenumbers/message/15873.

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."

Underbakke, D. "Record Sophie Germain Prime Pair." Jan 11, 2005. http://groups.yahoo.com/group/primenumbers/message/15872.

Underwood, P. "Sophie Germain Record Primes." Message to primeform e-mail list. May 3, 2006. http://groups.yahoo.com/group/primeform/message/7378.

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Weisstein, Eric W. "Sophie Germain Prime." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/SophieGermainPrime.html