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

merge' (x:xt) ys = x : (merge xt ys)

primes = ps ++ (diff ns $ foldr1 merge' $ map f $ tail primes)
    where ps  = [2,3,5]
          ns  = [7,9..]
          f p = [ m*p | m <- [p,p+2..]]

merge' 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.