# Prime numbers

## Prime Number Resources

In mathematics, a prime number (or a prime) is a natural number which has exactly two distinct natural number divisors: 1 and itself.

Prime Numbers at Wikipedia.

Sieve of Eratosthenes at Wikipedia.

HackageDB packages:

Numbers: An assortment of number theoretic functions.

NumberSieves: Number Theoretic Sieves: primes, factorization, and Euler's Totient.

primes: Efficient, purely functional generation of prime numbers.

## Finding Primes

### Simple Prime Sieve

The following is an elegant way to generate a list of all the prime numbers in the universe:

```
primes :: [Integer]
primes = sieve [2..]
where sieve (p:xs) = p : sieve [x | x<-xs, x `mod` p /= 0]
```

While appearing simple, this method is extremely inefficient and not recommended for more than a few 1000s of prime numbers, even when compiled. For every number it will test its divisibility by all prime numbers smaller than its smallest prime factor. This means that any prime will be checked for divisibility by all its preceding primes, when in fact just those not greater than its square root would suffice.

It in effect creates a nested linear structure of filters in front of the infinite numbers supply, and does so in extremely premature fashion, leading to a blowup of unnecessary filters. One way of fixing that would be to *postpone* the creation of filters until the right moment, as in

#### Postponed Filters Prime Sieve

```
primes :: [Integer]
primes = 2: 3: sieve (tail primes) [5,7..] where
sieve (p:ps) xs
= h ++ sieve ps [x|x<-t, x `rem` p /= 0] -- or: (filter ((/=0).(`rem`p)) t)
where (h,(_:t))=span (< p*p) xs
```

This only tests odd numbers, and by the least amount of primes for each numbers span between successive squares of primes.

Whereas the first version exhibits near O() behavior, this one exhibits near O() behavior, and is good for creating a few 100,000s of primes (compiled). It takes as much time to generate *100,000 primes* with it as it takes for the first one to generate *5500*, GHC-compiled.

Given an infinite list of prime numbers, we can implement primality tests and integer factorization:

```
isPrime n = n > 1 && n == head (primeFactors n)
primeFactors 1 = []
primeFactors n = go n primes
where
go n ps@(p:pt)
| p*p > n = [n]
| n `rem` p == 0 = p : go (n `quot` p) ps
| otherwise = go n pt
```

### Simple Prime Sieve II

The following method is slightly faster, creating *114,000 primes* in the same period of time. It works well for generating a few 100,000s of primes as well:

```
primes :: [Integer]
primes = 2:filter isPrime [3,5..]
where
isPrime n = all (not . divides n) $ takeWhile (\p -> p*p <= n) primes
divides n p = n `mod` p == 0
```

Instead of relying on nested filters, it tests by an explicit list of all the needed prime factors, but it recomputes this list, which will be the same for the increasing spans of numbers between the successive squares of primes.

### Simple Prime Sieve III

This is faster still, creating *158,000 primes* (again, GHC-compiled) in the same time span:

```
primes :: [Integer]
primes = 2:3:go 5 [] (tail primes)
where
divisibleBy d n = n `mod` d == 0
go start ds (p:ps) = let pSq = p*p in
foldr (\d -> filter (not . divisibleBy d)) [start, start + 2..pSq - 2] ds
++ go (pSq + 2) (p:ds) ps
```

This one explicitly maintains the list of primes needed for testing each odds span between successive primes squares, which it also explicitly generates. But it tests with nested `filter`

s, which it repeatedly recreates.

### Simple Prime Sieve IV

The needed list of prime factors for each range of odds is actually just the prefix of primes list itself and need not be generated at all. This code combines that idea with one-call testing by the explicit list of primes, and the direct generation of odds between the successive primes squares:

```
primes :: [Integer]
primes = 2: 3: sieve 0 primes' 5
primes' = tail primes
sieve k (p:ps) x
= [x | x<-[x,x+2..p*p-2], and [(x`rem`p)/=0 | p<-take k primes']]
++ sieve (k+1) ps (p*p+2)
```

It produces about *222,000 primes* in the same time span, and is good for creating about a million first primes, compiled.

The reason to have `sieve`

function available separately too is that it can also be used to produce primes above a given number, as in

```
primesFrom m = sieve (length h) ps $ m`div`2*2+1
where
(h,(_:ps)) = span (<= (floor.sqrt.fromIntegral) m) primes
```

It can thus produce a few primes e.g. above `239812076741689`

, which is a square of the millionth odd prime, without having to compute all the preceding primes (which would number in trillions).

### Simple Sieve of Eratosthenes

All the versions thus far try to *keep the primes* among all numbers by testing *each number* in isolation. Testing composites is cheap (as most of them will have small factors, so the test is soon aborted), but testing prime numbers is costly. The original sieve of Eratosthenes tries to *get rid of composites* by working on *spans* of numbers *at once* and thus gets its primes *"for free"*, as gaps between the generated/marked/crossed-off composites:

```
primes :: [Integer]
primes = 2: 3: 5: 7: sieve [(6,15)] (drop 2 primes) 11
where
sieve fs (p:ps) x = comb x comps
++ sieve ((2*p,q+2*p):fs') ps (q+2)
where
q = p*p
(streams,fs') = unzip (map gen fs)
gen (s,x) | x < q = (x:xs,f)
| True = ([],(s,x)) where (xs,f) = gen (s,x+s)
comps = foldl1 mrg streams
comb x (y:ys) | x==y = comb (x+2) ys
| True = x: comb (x+2) (y:ys)
comb x [] = if x==q then [] else [x]
mrg u@(a:t) w@(b:r) | a<b = a:t `mrg` w
| b<a = b:u `mrg` r
| True = a:t `mrg` r
mrg u w = if null u then w else u
```

This "marks" the odd composites in a given range by generating them - just as a person performing the original sieve would do, marking one by one the multiples of the relevant primes - and then combs through them looking for gaps.

Compared with the previous version, if compiled with GHC, it is steadily 1.2x slower and takes 3.5x more memory. But interpreted under both GHCi and WinHugs, it runs *2x to 4x faster*, takes *less* memory, and has better asymptotic behavior.

### Prime Wheels

The idea of only testing odd numbers can be extended further. For instance, it is a useful fact that every prime number other than 2 and 3 must be of the form or . Thus, we only need to test these numbers:

```
primes :: [Integer]
primes = 2:3:primes'
where
1:p:candidates = [6*k+r | k <- [0..], r <- [1,5]]
primes' = p : filter isPrime candidates
isPrime n = all (not . divides n) $ takeWhile (\p -> p*p <= n) primes'
divides n p = n `mod` p == 0
```

Here, `primes'`

is the list of primes greater than 3 and `isPrime`

does not test for divisibility by 2 or 3 because the `candidates`

by construction don't have these numbers as factors. We also need to exclude 1 from the candidates and mark the next one as prime to start the recursion.

Such a scheme to generate candidate numbers first that avoid a given set of primes as divisors is called a **prime wheel**. Imagine that you had a wheel of circumference 6 to be rolled along the number line. With spikes positioned 1 and 5 units around the circumference, rolling the wheel will prick holes exactly in those positions on the line whose numbers are not divisible by 2 and 3.

A wheel can be represented by its circumference and the spiked positions.

```
data Wheel = Wheel Integer [Integer]
```

We prick out numbers by rolling the wheel.

```
roll (Wheel n rs) = [n*k+r | k <- [0..], r <- rs]
```

The smallest wheel is the unit wheel with one spike, it will prick out every number.

```
w0 = Wheel 1 [1]
```

We can create a larger wheel by rolling a smaller wheel of circumference `n`

along a rim of circumference `p*n`

while excluding spike positions at multiples of `p`

.

```
nextSize (Wheel n rs) p =
Wheel (p*n) [r' | k <- [0..(p-1)], r <- rs, let r' = n*k+r, r' `mod` p /= 0]
```

Combining both, we can make wheels that prick out numbers that avoid a given list `ds`

of divisors.

```
mkWheel ds = foldl nextSize w0 ds
```

Now, we can generate prime numbers with a wheel that for instance avoids all multiples of 2, 3, 5 and 7.

```
primes :: [Integer]
primes = small ++ large
where
1:p:candidates = roll $ mkWheel small
small = [2,3,5,7]
large = p : filter isPrime candidates
isPrime n = all (not . divides n) $ takeWhile (\p -> p*p <= n) large
divides n p = n `mod` p == 0
```

It's a pretty big wheel with a circumference of 210 and allows us to calculate the first 10000 primes in convenient time.

A fixed size wheel is fine, but how about adapting the wheel size while generating prime numbers? See the functional pearl titled Lazy wheel sieves and spirals of primes for more.

### Implicit Heap

The following is a more efficient prime generator, implementing the sieve of Eratosthenes.

See also the message threads Re: "no-coding" functional data structures via lazyness for more about how merging ordered lists amounts to creating an implicit heap and Re: Code and Perf. Data for Prime Finders for an explanation of the `People a`

structure that makes it work when tying a knot.

```
data People a = VIP a (People a) | Crowd [a]
mergeP :: Ord a => People a -> People a -> People a
mergeP (VIP x xt) ys = VIP x $ mergeP xt ys
mergeP (Crowd xs) (Crowd ys) = Crowd $ merge xs ys
mergeP xs@(Crowd ~(x:xt)) ys@(VIP y yt) = case compare x y of
LT -> VIP x $ mergeP (Crowd xt) ys
EQ -> VIP x $ mergeP (Crowd xt) yt
GT -> VIP y $ mergeP xs yt
merge :: Ord a => [a] -> [a] -> [a]
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
foldTree :: (a -> a -> a) -> [a] -> a
foldTree f ~(x:xs) = f x . foldTree f . pairs $ xs
where pairs ~(x: ~(y:ys)) = f x y : pairs ys
primes, nonprimes :: [Integer]
primes = 2:3:diff [5,7..] nonprimes
nonprimes = serve . foldTree mergeP . map multiples $ tail primes
where
multiples p = vip [p*k | k <- [p,p+2..]]
vip (x:xs) = VIP x $ Crowd xs
serve (VIP x xs) = x:serve xs
serve (Crowd xs) = xs
```

`nonprimes`

effectively implements a heap, exploiting lazy evaluation.

### 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 -XBangPatterns #-}
module Primes (nthPrime) where
import Control.Monad.ST
import Data.Array.ST
import Data.Array.Base
import System
import Control.Monad
import Data.Bits
nthPrime :: Int -> Int
nthPrime 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
e <- unsafeRead a n
if e then
if n < cutoff then
let loop !j
| j < m = do
x <- unsafeRead a j
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 = nthPrime 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
```

### Using IntSet for a traditional sieve

```
module Sieve where
import qualified Data.IntSet as I
-- findNext - finds the next member of an IntSet.
findNext c is | I.member c is = c
| c > I.findMax is = error "Ooops. No next number in set."
| otherwise = findNext (c+1) is
-- mark - delete all multiples of n from n*n to the end of the set
mark n is = is I.\\ (I.fromAscList (takeWhile (<=end) (map (n*) [n..])))
where
end = I.findMax is
-- primes - gives all primes up to n
primes n = worker 2 (I.fromAscList [2..n])
where
worker x is
| (x*x) > n = is
| otherwise = worker (findNext (x+1) is) (mark x is)
```

## Testing Primality

### Miller-Rabin Primality Test

```
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)
```

## External links

- A collection of prime generators; the file "ONeillPrimes.hs" contains one of the fastest pure-Haskell prime generators. WARNING: Don't use the priority queue from that file for your projects: it's broken and works for primes only by a lucky chance.