A prime number
is a Sophie Germain prime
is also a prime.
The name originates from the fact that in 1823 Sophie Germain
proved that one subcase of the Fermat Last Theorem
holds for all the prime exponents such that is also prime.
It is conjectured that there are infinitely many Sophie Germain primes,
and that up to there are approximately
such primes, where
(in which the product is over all odd primes
is the so-called Hardy–Littlewood's twin prime constant.
Up to there are 1822848478 Sophie Germain primes,
and the formula above underestimates the actual number by about 5.4%.
The first Sophie Germain primes are
2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419 more terms
Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here