Is this a school science project or are you looking to push the envelope of number theory?
If it's for school, just try to limit the number of BigInteger operations per trial and time one iteration of your code well enough to set reasonable goals for your project. If you do a good job researching the work that has already been done and integrate that research into your search algorithm, you'll be in good shape.
If you are competing with the big players, you'll want to learn assembler and study the different algorithms for manipulating big numbers and the exact optimization techniques used by the manufacturer of your CPU chip. All this material is on the web. Try google.
I did a search for an odd perfect number in High School and kept my results as the exponents of the prime numbers starting with 3 because the theorems in that area worked with the exponents. You'll want to keep your numbers in a form compatible with the existing theorems on Mersenne primes.
BTW, I found this link:
http://www.utm.edu/research/primes/mersenne/index.html Good luck.