Prime numbers

primes :: [Integer]
  primes = sieve [2..]
    where sieve (p:xs) = p : sieve [x | x<-xs, x `mod` p /= 0]

isPrime n = n > 1 && n == head (primeFactors n)
 
  primeFactors 1 = []
  primeFactors n = go n primes
     where
     go n ps@(p:pt)
        | p*p > n        = [n]
        | n `rem` p == 0 = p : go (n `quot` p) ps
        | otherwise      = go n pt

primes :: [Integer]
  primes = 2:filter isPrime [3,5..]
     where
     isPrime n   = all (not . divides n) $ takeWhile (\p -> p*p <= n) primes
     divides n p = n `mod` p == 0

primes :: [Integer]
primes = 2:3:primes'
  where
    1:p:candidates = [6*k+r | k <- [0..], r <- [1,5]]
    primes'        = p : filter isPrime candidates
    isPrime n      = all (not . divides n) $ takeWhile (\p -> p*p <= n) primes'
    divides n p    = n `mod` p == 0

data Wheel = Wheel Int [Int]

roll (Wheel n rs) = [n*k+r | k <- [0..], r <- rs]

w0 = Wheel 1 [1]

nextSize (Wheel n rs) p =
  Wheel (p*n) [r' | k <- [0..(p-1)], r <- rs, let r' = n*k+r, r' `mod` p /= 0]

mkWheel ds = foldl nextSize w0 ds

primes :: [Integer]
primes = small ++ large
    where
    1:p:candidates = roll $ mkWheel small
    small          = [2,3,5,7]
    large          = p : filter isPrime candidates
    isPrime n      = all (not . divides n) $ takeWhile (\p -> p*p <= n) large
    divides n p    = n `mod` p == 0

data People a = VIP a (People a) | Crowd [a]
 
mergeP :: Ord a => People a -> People a -> People a
mergeP (VIP x xt) ys                    = VIP x $ mergeP xt ys
mergeP (Crowd xs) (Crowd ys)            = Crowd $ merge  xs ys
mergeP xs@(Crowd ~(x:xt)) ys@(VIP y yt) = case compare x y of
    LT -> VIP x $ mergeP (Crowd xt) ys
    EQ -> VIP x $ mergeP (Crowd xt) yt
    GT -> VIP y $ mergeP xs yt
 
 
merge :: Ord a => [a] -> [a] -> [a]
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
 
foldTree :: (a -> a -> a) -> [a] -> a
foldTree f ~(x:xs) = f x . foldTree f . pairs $ xs
    where pairs ~(x: ~(y:ys)) = f x y : pairs ys
 
primes, nonprimes :: [Integer]
primes    = 2:3:diff [5,7..] nonprimes
nonprimes = serve . foldTree mergeP . map multiples $ tail primes
    where
    multiples p = vip [p*k | k <- [p,p+2..]]
 
    vip (x:xs)       = VIP x $ Crowd xs
    serve (VIP x xs) = x:serve xs
    serve (Crowd xs) = xs

{-# OPTIONS -O2 -optc-O -XBangPatterns #-}
module Primes (nthPrime) where
 
import Control.Monad.ST
import Data.Array.ST
import Data.Array.Base
import System
import Control.Monad
import Data.Bits
 
nthPrime :: Int -> Int
nthPrime 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

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

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

find2km :: Integral a => a -> (a,a)
find2km n = f 0 n
    where 
        f k m
            | r == 1 = (k,m)
            | otherwise = f (k+1) q
            where (q,r) = quotRem m 2        
 
millerRabinPrimality :: Integer -> Integer -> Bool
millerRabinPrimality n a
    | a <= 1 || a >= n-1 = 
        error $ "millerRabinPrimality: a out of range (" 
              ++ show a ++ " for "++ show n ++ ")" 
    | n < 2 = False
    | even n = False
    | b0 == 1 || b0 == n' = True
    | otherwise = iter (tail b)
    where
        n' = n-1
        (k,m) = find2km n'
        b0 = powMod n a m
        b = take (fromIntegral k) $ iterate (squareMod n) b0
        iter [] = False
        iter (x:xs)
            | x == 1 = False
            | x == n' = True
            | otherwise = iter xs
 
pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
    where 
        f x n y
            | n == 1 = x `mul` y
            | r == 0 = f x2 q y
            | otherwise = f x2 q (x `mul` y)
            where
                (q,r) = quotRem n 2
                x2 = sq x
 
mulMod :: Integral a => a -> a -> a -> a
mulMod a b c = (b * c) `mod` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)

module Sieve where
import qualified Data.IntSet as I
 
-- findNext - finds the next member of an IntSet.
findNext c is | I.member c is = c
              | c > I.findMax is = error "Ooops. No next number in set."   
              | otherwise = findNext (c+1) is
 
-- mark - delete all multiples of n from n*n to the end of the set
mark n is = is I.\\ (I.fromAscList (takeWhile (<=end) (map (n*) [n..])))
                where
                    end = I.findMax is
 
-- primes - gives all primes up to n 
primes n = worker 2 (I.fromAscList [2..n])
                where
                    worker x is 
                     | (x*x) > n = is
                     | otherwise = worker (findNext (x+1) is) (mark x is)