A Sophie Germain Prime is a
prime number P for which 2P+1 is also a prime. These prime numbers include:
2, 3, 5, 11, 23,
29, 41, 53, 89, 113, 131...
It can be noted that a Sophie Germain prime p > 3
is of the form 6k−1, since adding one to them make them a multiple of six.
These primes were named after the French
mathematician MarieSophie Germain when she (around 1825) proved that
Fermat's
Last Theorem holds true for these primes. That is, if n is a Sophie Germain prime greater than 2, then there are no whole numbers
a, b, c such that
a^{n} + b^{n} = c^{n}.
It has been conjectured that there are infinitely
many Sophie Germain primes, however, this conjecture has not yet been proven.
The corresponding prime numbers of the
form 2P+1, where P is a Sophie Germain prime are called safe primes.
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