Safe 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 prime number $p$ is termed a safe prime if $p$ is odd and $(p-1)/2$ is also a prime number.

The corresponding prime $(p-1)/2$ is termed a Sophie Germain prime.

Relation with other properties

Related properties of primes or pairs of primes

• Sophie Germain prime is a prime $q$ such that $2q+1$ is also a prime.
• Twin primes are primes that differ by two.

Related properties of longer chains of primes

• Cunningham chain is a chain of primes $q_{1}<q_{2}<\dots <q_{k}$ such that $q_{i+1}=2q_{i}+1$.
• Bitwin chain is a collection of multiple pairs of twin primes, each pair being double the previous one.

Facts

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