A Sophie Germain prime is a prime 
for which 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 Oct. 2026 is given by 
where
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(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.
For a Sophie Germain prime 
of the form 
, the prime 
is a proper divisor of the
Mersenne number 
, so 
is composite (Dubner
1996, p. 393).
Sophie Germain proved around 1825 that the first case of Fermat's last theorem is true for such primes (Dubner 1996, p. 393), 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
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