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, ... (OEIS
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, ... (OEIS
A092816).
The largest known proven Sophie Germain prime pair as of Feb. 29, 2016 is given by
where
(Caldwell), each of which has decimal digits (PrimeGrid).
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.
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
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, ... (OEIS
A005384). It is not known if there are an infinite
number of Sophie Germain primes (Hoffman 1998, p. 190).
for 
, 2, ... are 3, 10, 37, 190, 1171, 7746, 56032, ... (OEIS
A092816).
where
decimal digits (PrimeGrid).
of the form 
correspond to the indices of composite Mersenne
numbers 
.
is a Sophie Germain prime, then there do not exist integers 
, 
, and 
different from 0 and none a multiple of 
such that
