# Prime numbers

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## Revision as of 13:24, 15 July 2007

The following is an elegant (and highly inefficient) way to generate a list of all the prime numbers in the universe:

primes = sieve [2..] where sieve (p:xs) = p : sieve (filter (\x -> x `mod` p > 0) xs)

With this definition made, a few other useful (??) functions can be added:

is_prime n = n `elem` (takeWhile (n >=) primes) factors n = filter (\p -> n `mod` p == 0) primes factorise 1 = [] factorise n = let f = head $ factors n in f : factorise (n `div` f)

takeWhile

The following is a more efficient prime generator, implementing the sieve of Eratosthenes:

merge xs@(x:xt) ys@(y:yt) = case compare x y of LT -> x : (merge xt ys) EQ -> x : (merge xt yt) GT -> y : (merge xs yt) diff xs@(x:xt) ys@(y:yt) = case compare x y of LT -> x : (diff xt ys) EQ -> diff xt yt GT -> diff xs yt primes, nonprimes :: [Int] primes = [2,3,5] ++ (diff [7,9..] nonprimes) nonprimes = foldr1 f . map g $ tail primes where f (x:xt) ys = x : (merge xt ys) g p = [ n*p | n <- [p,p+2..]]

nonprimes

ShowS

to avoid explicitly coding efficient concatenable strings. This is generalized by the DList package.

## Bitwise prime sieve

Count the number of prime below a given 'n'. Shows fast bitwise arrays, and an example of Template Haskell to defeat your enemies.

{-# OPTIONS -O2 -optc-O -fbang-patterns #-} module Primes (pureSieve) where import Control.Monad.ST import Data.Array.ST import Data.Array.Base import System import Control.Monad import Data.Bits pureSieve :: Int -> Int pureSieve n = runST ( sieve n ) sieve n = do a <- newArray (3,n) True :: ST s (STUArray s Int Bool) let cutoff = truncate (sqrt (fromIntegral n)) + 1 go a n cutoff 3 1 go !a !m cutoff !n !c | n >= m = return c | otherwise = do e <- unsafeRead a n if e then if n < cutoff then let loop !j | j < m = do x <- unsafeRead a j when x $ unsafeWrite a j False loop (j+n) | otherwise = go a m cutoff (n+2) (c+1) in loop ( if n < 46340 then n * n else n `shiftL` 1) else go a m cutoff (n+2) (c+1) else go a m cutoff (n+2) c

And places in a module:

{-# OPTIONS -fth #-} import Primes main = print $( let x = pureSieve 10000000 in [| x |] )

Run as:

$ ghc --make -o primes Main.hs $ time ./primes 664579 ./primes 0.00s user 0.01s system 228% cpu 0.003 total