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    <p>@DrPizza Profiling only really helps improve an implementation, it doesn't reveal opportunities for parallel execution, or suggest better algorithms (unless you've experience to the otherwise, in which case I'd really like to see your profiler).</p> <p>I've only single core machines at home, but ran a Java equivalent of your BitArray sieve, and a single threaded version of the inversion of the sieve - holding the marking primes in an array, and using a <a href="http://en.wikipedia.org/wiki/Wheel_factorization" rel="nofollow noreferrer">wheel</a> to reduce the search space by a factor of five, then marking a bit array in increments of the wheel using each marking prime. It also reduces storage to O(sqrt(N)) instead of O(N), which helps both in terms of the largest N, paging, and bandwidth. </p> <p>For medium values of N (1e8 to 1e12), the primes up to sqrt(N) can be found quite quickly, and after that you should be able to parallelise the subsequent search on the CPU quite easily. On my single core machine, the wheel approach finds primes up to 1e9 in 28s, whereas your sieve (after moving the sqrt out of the loop) takes 86s - the improvement is due to the wheel; the inversion means you can handle N larger than 2^32 but makes it slower. Code can be found <a href="http://www.tincancamera.com/examples/java/primes/" rel="nofollow noreferrer">here</a>. You could parallelise the output of the results from the naive sieve after you go past sqrt(N) too, as the bit array is not modified after that point; but once you are dealing with N large enough for it to matter the array size is too big for ints.</p>
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