Efficient storage of prime numbers

Question

For a library, I need to store the first primes numbers up to a limit L. This collection must have a O(1) lookup time (to check whether a number is prime or not) and it must

Answer 1

An alternative to packed bitmaps and wheels - but equally efficient in certain contexts - is storing the differences between consecutive primes. If you leave out the number 2 as usual then all differences are even. Storing difference/2 you can get up to 2^40ish regions (just before 1999066711391) using byte-sized variables.

The primes up 2^32 require only 194 MByte, compared to 256 MByte for an odds-only packed bitmap. Iterating over delta-stored primes is much faster than for wheeled storage, which includes the modulo-2 wheel known as odds-only bitmap.

For ranges from 1999066711391 onwards, bigger cell size or variable-length storage are needed. The latter can be extremely efficient even if very simple schemes are used (e.g. keep adding until a byte < 255 has been added, like in LZ4-style compression), because of the extremely low frequency of gaps longer than 510/2.

For efficiency's sake it is best to divide the range into sections (pages) and manage them B-Tree style.

Entropy-coding the differences (Huffmann or arithmetic coding) cuts permanent storage requirements to a bit less than half, which is close to the theoretical optimum and better than lists or wheels compressed using the best available packers.

If the data is stored uncompressed then it is still much more compact than files of binary or textual numbers, by an order of magnitude or more. With a B-Tree style index in place it is easy to simply map sections into memory as needed and iterate over them at blazing speed.