Sophie Germain Prime

A Prime is said to be a Sophie Germain prime if both and are Prime. The first few Sophie Germainprimes 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 is a Sophie Germain prime, there do not exist Integers , , and different from 0 and not multiplesof such that

Sophie Germain primes of the form (which makes a Prime) correspond to theindices of composite Mersenne Numbers . Since the largest known CompositeMersenne Number is with , is the largest known Sophie Germain prime.

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. SequenceA005384/M0731in ``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.

• Recurrence Relation
• Recurrence Sequence
• Recurring Digital Invariant
• Recursion
• Recursive Function
• Recursive Monotone Stable Quadrature
• Red-Black Tree
• Red Net
• Reduced Amicable Pair
• Reduced Fraction
• Reduced Latitude
• Reducible Crossing
• Reducible Matrix
• Reducible Representation
• Reduction of Order
• Reduction Theorem
• Redundancy
• Reeb Foliation
• Reef Knot
• Re-Entrant Circuit
• Refinement
• Reflection
• Reflection Property
• Reflection Relation
• Reflexible