# 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
```

## Yet another set of prime functions

I'm sure this isn't the most efficient, but it is good enough for many applications and it is simple enough for a beginning Haskeller like me to create. It only checks prime numbers as factors when trying to factor a new candidate and only checks odd candidates, otherwise it is pretty straightforward. Working more on efficiency wasn't worth it for the program that used it. Uses integral instead of int to allow for bigger numbers. Thanks to this, I now know that none of my (10-digit) phone numbers are prime numbers. Now to find the factors of my phone numbers...

```--| Simple prime functions
--| Produces an infinite list of primes.
myprimelist :: Integral a => [a]
myprimelist = 2 : [x | x <- [3,5..], myprimecheck x (takeWhile (<x) myprimelist)]

--| simple nth prime using the list
mynthprime :: (Integral a, Integral b) => a -> b
mynthprime 1 = 2
mynthprime n = head \$ drop  (fromIntegral (n-1)) myprimelist

--| Check a number for primeness.
myisprime :: Integral a => a -> Bool
myisprime n | n<2 = False
| n<4 = True
| even n = False
| otherwise = myprimecheck n (takeWhile (<n) myprimelist)

--| Used by myisprime and myprimelist to check for primeness.
myprimecheck :: Integral a => a -> [a] -> Bool
myprimecheck n [] = True
myprimecheck n (f:fs)	= if f*f <= n
then
if (n `mod` f)==0
then False
else myprimecheck n fs
else True
```

## Yet another test prime functions

```find2km :: Integral a => a -> (a,a)
find2km n = f 0 n
where
f k m
| r == 1 = (k,m)
| otherwise = f (k+1) q
where (q,r) = quotRem m 2

millerRabinPrimality :: Integer -> Integer -> Bool
millerRabinPrimality n a
| a <= 1 || a >= n-1 =
error \$ "millerRabinPrimality: a out of range ("
++ show a ++ " for "++ show n ++ ")"
| n < 2 = False
| even n = False
| b0 == 1 || b0 == n' = True
| otherwise = iter (tail b)
where
n' = n-1
(k,m) = find2km n'
b0 = powMod n a m
b = take (fromIntegral k) \$ iterate (squareMod n) b0
iter [] = False
iter (x:xs)
| x == 1 = False
| x == n' = True
| otherwise = iter xs

pow' :: (Num a, Integral b) => (a -> a -> a) -> (a -> a) -> a -> b -> a
pow' _ _ _ 0 = 1
pow' mul sq x' n' = f x' n' 1
where
f x n y
| n == 1 = x `mul` y
| r == 0 = f x2 q y
| otherwise = f x2 q (x `mul` y)
where
(q,r) = quotRem n 2
x2 = sq x

mulMod :: Integral a => a -> a -> a -> a
mulMod a b c = (b * c) `mod` a
squareMod :: Integral a => a -> a -> a
squareMod a b = (b * b) `rem` a
powMod :: Integral a => a -> a -> a -> a
powMod m = pow' (mulMod m) (squareMod m)
```