# Difference between revisions of "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.