The following Mersenne primes are known.
The way to determine if is prime is to use the Lucas-Lehmer test:
Lucas_Lehmer_Test(p): u := 4 for i from 3 to p do u := u^2-2 mod 2^p-1 od if u == 0 then 2^p-1 is prime else 2^p-1 is composite fi
All exponents less than 1,481,800 have now been tested at least once.
An introduction to the theory of numbers. G.H. Hardy, E.M. Wright. Fifth edition, 1979, Oxford.