Prime numbers

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)

(Note the use of takeWhile to prevent the infinite list of primes requiring an infinite amount of CPU time and RAM to process!)

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 effectively implements a heap, exploiting Haskell's lazy evaluation model. For another example of this idiom see the Prelude's ShowS type, which again exploits Haskell's lazy evaluation model to avoid explicitly coding efficient concatenable strings. This is generalized by the DList package.