Mersenne numbers are very special because we have a VERY efficient method for proving one is prime : the LLT. Moreover, the more Mersenne primes we find, the more we learn n about them, leading for possible new knowledge. Moreover, the very complex programs used for checking these numbers help to check fundamental HW and SW tools.
I helped the GIMPS to check 7 of the Mersenne primes they found.
I found : Mq = 2^q-1 = (3x)^2-(3qy)^2.