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.

Occurrence

Initial values

The ID of the sequence in the Online Encyclopedia of Integer Sequences is A005385

The first few safe primes are: 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207, 2447, 2459, 2579, 2819, 2879, 2903.

The first few primes that are not safe primes are: 2, 3, 13, 17, 19.

Density in primes

Cutoff $n$	Number of primes $\leq n$	Number of safe primes $\leq n$	Proportion of primes that are safe primes
10	4	2	$1 / 2 = 0.5$
100	25	7	$7 / 25 = 0.28$
1000	168	24	$25 / 168 \approx 0.15$

Relation with other properties

Related properties of primes or pairs of primes

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

Related properties of longer chains of primes

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

Facts