arithmoi-0.4.1.3: Efficient basic number-theoretic functions. Primes, powers, integer logarithms.

Copyright (c) 2011 Daniel Fischer MIT Daniel Fischer Provisional Non-portable (GHC extensions) None Haskell2010

Math.NumberTheory.Primes.Sieve

Contents

Description

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.

Synopsis

# Limitations

There are three factors limiting the range of these sieves.

1. Memory
2. Overflow
3. 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(sqrt n/log n)` space.We store `16` bytes of information per prime, thus a Gigabyte of memory takes you to about `1.6*10^18`. The `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 `10^37`.

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 `primeSieve` to shortly above `8*10^9`, the latter limits the range of the segmented sieves to about `1.7*10^18`.

# Sieves and lists

primes :: [Integer] Source

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 `1.7*10^18`.

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

data PrimeSieve Source

Compact store of primality flags.

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.

List of primes in the form of a list of `PrimeSieve`s, 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 n` creates the list of `PrimeSieve`s 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.

Generate a list of primes for consumption from a `PrimeSieve`.