Making a very large Java array

Depending on how you need to access the array, you might find a RandomAccessFile will allow you to use a file which is larger than will fit in memory. However, the performance you get is very dependant on your access behaviour.

I wrote a version of the Sieve of Eratosthenes for Project Euler which worked on chunks of the search space at a time. It processes the first 1M integers (for example), but keeps each prime number it finds in a table. After you've iterated over all the primes found so far, the array is re-initialised and the primes found already are used to mark the array before looking for the next one.

The table maps a prime to its 'offset' from the start of the array for the next processing iteration.

This is similar in concept (if not in implementation) to the way functional programming languages perform lazy evaluation of lists (although in larger steps). Allocating all the memory up-front isn't necessary, since you're only interested in the parts of the array that pass your test for primeness. Keeping the non-primes hanging around isn't useful to you.

This method also provides memoisation for later iterations over prime numbers. It's faster than scanning your sparse sieve data structure looking for the ones every time.