# 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.

## 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 -fbang-patterns #-}

module Primes (pureSieve) where

import Data.Array.ST
import Data.Array.Base
import System
import Data.Bits

pureSieve :: Int -> Int
pureSieve 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
if e then
if n < cutoff
then let loop !j
| j < m     = do
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 = pureSieve 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
```