Sophie Germain Prime

A Prime $p$ is said to be a Sophie Germain prime if both $p$ and $2p+1$ are Prime. The first few Sophie Germain primes are 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, ... (Sloane's A005384).

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, there do not exist Integers $x$, $y$, and $z$ different from 0 and not multiples of $p$ such that

$$x^p+y^p=z^p.$$

Sophie Germain primes $p$ of the form $p=k\cdot 2^n-1$ (which makes $2p+1$ a Prime) correspond to the indices of composite Mersenne Numbers $M_p$. Since the largest known Composite Mersenne Number is $M_p$ with $p=39051\times 2^{6001}-1$, $p$ is the largest known Sophie Germain prime.

See also Cunningham Chain, Fermat's Last Theorem, Mersenne Number, Twin Primes

References

Dubner, H. ``Large Sophie Germain Primes.'' Math. Comput. 65, 393-396, 1996.

Ribenboim, P. ``Sophie Germane 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. Sequence A005384/M0731 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.