## You are here

Homesafe prime

## Primary tabs

# safe prime

Given a prime $q$, if ${{q-1}\over 2}=p$, where $p$ is another prime number, then $q$ is a safe prime. It follows that $q-1=2p$ (and thus a semiprime) and $p$ is a Sophie Germain prime.

Every safe prime matches to a Sophie Germain prime this way, and some safe primes are Sophie Germain primes themselves too, matching to yet another safe prime. A set of primes linked in this way is called a Cunningham chain of the first kind. For example, 2, 5, 11, 23, 47. With Pocklington’s criterion, one can prove the primality of a safe prime if one has first proven the primality of its matching Sophie Germain prime.

The first few safe primes are 5, 7, 11, 23, 47, 59, 83, 107, 167, 179, 227, … These are listed in A005385 of Sloane’s OEIS. As of 2010, the largest known safe prime is $183027\times 2^{{265441}}-1$.

Safe primes have been used in various methods of cryptography, but the safety of their use depends not just on these mathematical properties but also on their being large enough that their multiples can’t be factored in a reasonable period of time by contemporary computers.

# 0.1 External links

## Mathematics Subject Classification

11A41*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: Prime numbers out of sequence by Rubens373

Oct 7

new question: Lorenz system by David Bankom

Oct 19

new correction: examples and OEIS sequences by fizzie

Oct 13

new correction: Define Galois correspondence by porton

Oct 7

new correction: Closure properties on languages: DCFL not closed under reversal by babou

new correction: DCFLs are not closed under reversal by petey

Oct 2

new correction: Many corrections by Smarandache

Sep 28

new question: how to contest an entry? by zorba

new question: simple question by parag