|Portability||Non-portable (GHC extensions)|
|Maintainer||Daniel Fischer <firstname.lastname@example.org>|
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
O(sqrt n/log n) space.
There are three factors limiting the range of these sieves.
- The internal representation of the state
An Eratosthenes type sieve needs to store the primes up to the square root of
the currently sieved region, thus requires
O(n/log n) space.We store
of information per prime, thus a Gigabyte of memory takes you to about
log doesn't change much in that range, so as a first approximation, doubling
the storage increases the sieve range by a factor of four.
On a 64-bit system, this is (currently) the only limitation to be concerned with, but
with more than four Terabyte of memory, the fact that the internal representation
currently limits the sieve range to about
6.8*10^25 could become relevant.
Overflow in array indexing doesn't become a concern before memory and internal
representation would allow to sieve past
On a 32-bit system, the internal representation imposes no additional limits,
but overflow has to be reckoned with. On the one hand, the fact that arrays are
Int-indexed restricts the size of the prime store, on the other hand, overflow
in calculating the indices to cross off multiples is possible before running out
of memory. The former limits the upper bound of the monolithic
8*10^9, the latter limits the range of the segmented sieves to
List of primes.
Since the sieve uses unboxed arrays, overflow occurs at some point.
On 64-bit systems, that point is beyond the memory limits, on
32-bit systems, it is at about
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
, that corresponds to an
upper bound near
32-bit systems, that
is often within memory limits, so don't give bounds larger than