# Prime numbers

### From HaskellWiki

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.