Safe prime: Difference between revisions

Revision as of 22:13, 21 April 2009

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 $2 q + 1$ 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 $q_{i + 1} = 2 q_{i} + 1$ .
Bitwin chain is a collection of multiple pairs of twin primes, each pair being double the previous one.

Facts

@@ Line 6: / Line 6: @@
 The corresponding prime <math>(p-1)/2</math> is termed a [[defining ingredient::Sophie Germain prime]].
+==Relation with other properties==
+===Related properties of primes or pairs of primes===
+* [[Sophie Germain prime]] is a prime <math>q</math> such that <math>2q + 1</math> 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 <math>q_1 < q_2 < \dots < q_k</math> such that <math>q_{i+1} = 2q_i + 1</math>.
+* [[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]]