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.
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Definition

A Sophie Germain prime is a prime number $p$ such that $2p+1$ is also prime. The corresponding prime $2p+1$ is termed a safe prime.

Occurrence

Initial examples

The first few Sophie Germain primes are $2,3,5,11,23,29$. The first prime that is not a Sophie Germain prime is $7$.

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 $n$ is $O(n/{\log }^{2}n)$, in other words, there exist constants $a,b$ such that the following holds for all sufficiently large $n$:

$\frac{an}{{\log }^{2}n}\le {\pi }_{SG}(n)\le \frac{bn}{{\log }^{2}n}$.

where ${\pi }_{SG}(n)$ denotes the number of Sophie Germain primes less than or equal to $n$.

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 $1$ modulo $3$, 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