Prime number

http://en.wikipedia.org/wiki/Prime_number

There are many particular types of primes:

>> Primes of the form 2^p − 1, where p is a prime number, are known as Mersenne primes.

>> Primes of the form 2^{2^k} + 1 are Fermat primes.

>> Prime numbers p where 2p + 1 is also prime are known as Sophie Germain primes.

I'm particularly interested in prime numbers of the forms (4k + 1) and (4k + 3).

Primes of the form (4k + 1) can be expressed as the sum of two squares, and prime numbers of the form (4k + 3) can be expressed as the sum of four squares.

We can find a number, a, such that a^2 = -1 (mod p), where p is a prime of the form 4k+1

And, I'm also interested in prime numbers p such that exactly one of the numbers 2p-1 and 2^p - 1 is prime.