Mersenne Prime 

Mersenne numbers which are also prime are called Mersenne Primes.
Most of the largest primes known are Mersenne Primes as there exists an efficient method of testing the primality of such numbers called the LucasLehmer test.
There exists an International Project to find unknown Mersenne primes called the Great Internet Mersenne Prime Search (GIMPS) where volunteers allow the downtime on their computers to be used for the endeavour.
$M_p=2^p1$ is known to be prime for 47 values of p = 2, 3, 5, 7, ... , 243112609
$2^{243112609}1$ is the largest known prime number, was discovered in August 2008 and is 12,978,189 digits long.
Enrichment Task
Show that if n (=ab) is a composite number then $2^a1$ is a factor of $2^n1$ therefore if $2^n1$ is prime then n is prime.
See also: Fermat prime.