Prime numbers
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1 Simple Prime Sieve
The following is an elegant (and highly inefficient) way to generate a list of all the prime numbers in the universe:
primes :: [Integer] primes = sieve [2..] where sieve (p:xs) = p : sieve [x | x<-xs, x `mod` p /= 0]
Given an infinite list of prime numbers, we can implement primality tests and integer factorization:
isPrime n = n > 1 && n == head (factorize n) primeFactors 1 = [] primeFactors n = go n primes where go n ps@(p:pt) | p*p > n = [n] | x `rem` p == 0 = p : go (n `quot` p) ps | otherwise = go n pt
2 Simple Prime Sieve II
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
3 Prime Wheels
Notice in the above Prime Sieve II, only odd numbers are tested, because we know that all the even numbers (greater than 2) are composite. In effect, odd numbers, and not even numbers, are candidates for primality testing.
A prime wheel is a scheme to generate candidate numbers that are "pre-screened" so that they don't have certain predetermined divisors. For example, suppose we want candidates that are neither even nor divisible by 3. In that case, we need numbers of the form 6n + {1,5}.
primes :: [Integer] primes = 2:3:5:filter isPrime wheel where -- these numbers are automatically not divisible by 2 or 3 wheel = 7:11:map (6+) wheel -- don't bother to check for divisibility by 2 or 3 ps = drop 2 primes isPrime n = all (not . divides n) $ takeWhile (\p -> p*p <= n) ps divides n p = n `mod` p == 0
This generator runs slightly faster than Prime Sieve II above because it doesn't bother to perform prime testing on multiples of 2 or 3.
Here is why the scheme is called a prime wheel. Imagine that you had a wheel of circumference 6, and you are rolling that wheel along the number line. The wheel is marked along the edges to automatically tell you which numbers are candidates and which numbers to exclude. Specifically, multiples of 2, 3 or 6 are excluded, while numbers of the form 6n+1 and 6n+5 are candidates.
We can go further and exclude multiples of 5. To exclude multiples of 2, 3, and 5, our wheel has to increase in multiples of 30.
primes :: [Integer] primes = 2:3:5:7:11:13:17:19:23:29:filter isPrime wheel where -- these numbers are automatically not divisible by 2, 3, or 5 wheel = 31:37:41:43:47:49:53:59:map (30+) wheel -- don't bother to check for divisibility by 2, 3, or 5 ps = drop 3 primes isPrime n = all (not . divides n) $ takeWhile (\p -> p*p <= n) ps divides n p = n `mod` p == 0
This generator runs slightly faster than the (2,3) prime wheel because it doesn't bother to check multiples of 2, 3, or 5.
We can go even further and exclude multiples of 7, but this requires a much bigger wheel, and it provides only a very small additional speed-up. This wheel has a length of 210, and at this point we are probably well beyond the point of diminishing returns.
primes :: [Integer] primes = initPrimes ++ filter isPrime wheel where initPrimes = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47, 53,59,61,67,71,73,79,83,89,97, 101,103,107,109,113,127,131,137,139,149, 151,157,163,167,173,179,181,191,193,197,199] -- the following numbers are automatically not divisible by 2, 3, 5, or 7 wheel = [211,221,223,227,229,233,239,241,247,251,253,257,263,269, 271,277,281,283,289,293,299,307,311,313,317,319,323,331, 337,341,347,349,353,359,361,367,373,377,379,383,389,391, 397,401,403,407,409,419] ++ map (210+) wheel -- don't bother to check for divisibility by 2, 3, 5, or 7 ps = drop 4 primes isPrime n = all (not . divides n) $ takeWhile (\p -> p*p <= n) ps divides n p = n `mod` p == 0
4 Implicit Heap
The following is a more efficient prime generator, implementing the sieve of Eratosthenes.
See also the message threads Re: "no-coding" functional data structures via lazyness for more about how merging ordered lists amounts to creating an implicit heap and Re: Code and Perf. Data for Prime Finders for an explanation of the People a structure that makes it work when tying a knot.
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
nonprimes effectively implements a heap, exploiting lazy evaluation.
5 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 -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
And places in a module:
{-# OPTIONS -fth #-} import Primes main = print $( let x = nthPrime 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
6 Miller-Rabin Primality Test
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)