What is the current status on Mersenne primes?

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.

References

An introduction to the theory of numbers. G.H. Hardy, E.M. Wright. Fifth edition, 1979, Oxford.