Further Properties
There is no special primality test for safe primes the way there is for Fermat primes and Mersenne primes. However, Pocklington's criterion can be used to prove the primality of 2p+1 once one has proven the primality of p.
With the exception of 5, there are no Fermat primes that are also safe primes. Since Fermat primes are of the form F = 2n + 1, it follows that (F − 1)/2 is a power of two.
With the exception of 7, there are no Mersenne primes that are also safe primes. This follows from the statement above that all safe primes except 7 are of the form 6k − 1. Mersenne primes are of the form 2m − 1, but 2m − 1 = 6k − 1 would imply that 2m is divisible by 6, which is impossible.
Just as every term except the last one of a Cunningham chain of the first kind is a Sophie Germain prime, so every term except the first of such a chain is a safe prime. Safe primes ending in 7, that is, of the form 10n + 7, are the last terms in such chains when they occur, since 2(10n + 7) + 1 = 20n + 15 is divisible by 5.
If a safe prime q is congruent to 7 modulo 8, then it is a divisor of the Mersenne number with its matching Sophie Germain prime as exponent.
Read more about this topic: Safe Prime
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