arithmoi- Efficient basic number-theoretic functions. Primes, powers, integer logarithms.

PortabilityNon-portable (GHC extensions)
MaintainerDaniel Fischer <>



Prime generation using a sieve. Currently, an enhanced sieve of Eratosthenes is used, switching to an Atkin sieve is planned (if I get around to implementing it and it's not slower).

The sieve used is segmented, with a chunk size chosen to give good (enough) cache locality while still getting something substantial done per chunk. However, that means we must store data for primes up to the square root of where sieving is done, thus sieving primes up to n requires O(sqrt n/log n) space.



primes :: [Integer]Source

List of primes. Since the sieve uses unboxed arrays, overflow occurs at some point, but not before 10^6*fromIntegral (maxBound :: Int) (I forgot where exactly).

sieveFrom :: Integer -> [Integer]Source

sieveFrom n creates the list of primes not less than n.

data PrimeSieve Source

Compact store of primality flags.

primeSieve :: Integer -> PrimeSieveSource

Sieve primes up to (and including) a bound. For small enough bounds, this is more efficient than using the segmented sieve.

Since arrays are Int-indexed, overflow occurs when the sieve size comes near maxBound :: Int, that corresponds to an upper bound near 15/8*maxBound. On 32-bit systems, that is often within memory limits, so don't give bounds larger than 8*10^9 there.

psieveList :: [PrimeSieve]Source

List of primes in the form of a list of PrimeSieves, more compact than primes, thus it may be better to use psieveList >>= primeList than keeping the list of primes alive during the entire run.

psieveFrom :: Integer -> [PrimeSieve]Source

psieveFrom n creates the list of PrimeSieves starting roughly at n. Due to the organisation of the sieve, the list may contain a few primes less than n. This form uses less memory than [Integer], hence it may be preferable to use this if it is to be reused.

primeList :: PrimeSieve -> [Integer]Source

Generate a list of primes for consumption from a PrimeSieve.