Sophie Germain prime

This article defines a property that can be evaluated for a prime number. In other words, every prime number either satisfies this property or does not satisfy this property.
View other properties of prime numbers | View other properties of natural numbers

Definition

A Sophie Germain prime is a prime number such that is also prime. The corresponding prime is termed a safe prime.

Occurrence

Initial values

The first few Sophie Germain primes are:

2, 3, 5, 11, 23, 29, 41, 53, 83, 89, [SHOW MORE]

113, 131, 173, 179, 191, 233, 239, 251, 281, 293, 359, 419, 431, 443, 491, 509, 593, 641, 653, 659, 683, 719, 743, 761, 809, 911, 953, 1013, 1019, 1031, 1049, 1103, 1223, 1229, 1289, 1409, 1439, 1451, 1481, 1499, 1511, 1559

View list on OEIS

The first few primes that are not Sophie Germain primes are: 7, 13, 17, 19, 31.

Density in primes

Cutoff	Number of primes	Number of Sophie Germain primes	Proportion of primes that are Sophie Germain primes	Number of Sophie Germain primes divided by
10	4	3
100	25	10
1000	168	37

Infinitude conjecture

Further information: Infinitude conjecture for Sophie Germain primes

It is conjectured that there are infinitely many Sophie Germain primes (or equivalently, that there are infinitely many safe primes). It is also conjectured that the number of Sophie Germain primes less than or equal to is , in other words, there exist constants such that the following holds for all sufficiently large :

.

where denotes the number of Sophie Germain primes less than or equal to .

Other related conjectures to the existence of Sophie Germain primes are:

• Conjecture on existence of Cunningham chains of the first kind of arbitrary length
• Conjecture on existence of bitwin chains of arbitrary length

Infinitude of complement

Most primes are not Sophie Germain primes. It is rather easy to see that the number of primes that are not Sophie Germain primes is infinite: for instance, there are infinitely many primes that are modulo , and none of them can be a Sophie Germain prime.

Facts

While there are not too many interesting facts about the structure of Sophie Germain primes per se, the structure of the corresponding safe primes is very interesting:

• Quadratic nonresidue that is not minus one is primitive root for safe prime
• Safe prime has plus or minus two as primitive root

Testing

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A005384