Prime numbers

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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. The smallest prime is thus 2.

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

Any natural number is representable as a product of powers of its prime factors, and so a prime has no prime divisors other then itself. That means that starting with 2, for each newly found prime we can eliminate from the rest of the numbers all such numbers that have this prime as their divisor, giving us the next available number as next prime. This is known as sieving the natural numbers, so that in the end what we are left with are just primes.

The Classic Simple Primes Sieve

Attributed to David Turner (SASL Language Manual, 1975), the following is a direct translation of that idea, generating a list of all prime numbers:

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

This should only be considered a specification, not a code. When run as is, it is extremely inefficient because it starts up the filters prematurely, immediately after each prime, instead of only after the prime's square has been reached. To be admitted as prime, each number will be tested for divisibility here by all its preceding primes, while just those not greater than its square root would suffice. This means that e.g. to find the 1001st prime (7927), 1000 filters are used, when in fact just the first 24 are needed (upto 89's filter only).

So this in effect creates a cascade of nested filters in front of the infinite numbers supply, and in extremely premature fashion at that. One way of fixing that would be to postpone the creation of filters until the right moment, by decoupling the primes supply from the numbers supply, as in

Postponed Filters Sieve

  primes :: [Integer]
  primes = 2: 3: sieve (tail primes) [5,7..]
    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 can be seen as essential framework for all the code to come. It only tests odd numbers, and only by the primes that are needed, for each numbers span between successive squares of primes. To find the 1001st prime, the divisibility test is performed by only 24 nested filters corresponding to the first 24 odd primes.

Whereas the first version exhibits near O() behavior, this one exhibits near O() behavior, with an orders-of-magnitude speedup.

There is another way for the composites to be found - by generating all the multiples of successive primes, in advance. Any number thus generated will of course be divisible by the corresponding prime.

Postponed Multiples Removal i.e. Euler's Sieve

Instead of testing each number for divisibility by a prime we can just remove the prime's multiples in advance. We gain in speed because we now get the primes for free, after all the multiples are removed on a particular span, without performing any divisibility tests at all:

  primes :: [Integer]
  primes = 2: 3: sieve (tail primes) [5,7..]  
    sieve (p:ps) xs = h ++ sieve ps (t `minus` tail [q,q+2*p..])
                      where (h,~(_:t)) = span (< q) xs 
                            q          = p*p

  minus :: (Ord a) => [a] -> [a] -> [a]
  minus a@(x:xs) b@(y:ys) = case compare x y of 
                              LT -> x: xs `minus` b
                              EQ ->    xs `minus` ys
                              GT ->    a  `minus` ys
  minus a        b        = a

This is, in fact, Euler's sieve.

Merged Multiples Removal Sieve

(...((s-a)-b)-...) is the same as (s-(a+b+...)), and so we can just remove the merged infinite primes multiples, each starting at its prime's square, from the initial numbers supply. This way we needn't explicitly jump to a prime's square because it's where its multiples start anyway:

  primes :: [Integer]
  primes = 2:primes'
    primes' = [3] ++ [5,7..] `minus` foldr merge' [] mults
    mults   = map (\p->let q=p*p in (q,tail [q,q+2*p..])) $ primes'
    merge' (q,qs) xs = q : merge qs xs

  merge :: (Ord a) => [a] -> [a] -> [a]  
  merge a@(x:xs) b@(y:ys) = case compare x y of 
                              LT -> x: merge xs b 
                              EQ -> x: merge xs ys
                              GT -> y: merge a  ys
  merge a        b        = if null a then b else a

This code is yet faster. Its main deficiency still is that it creates a linear nested merging structure, instead of a tree-like structure. Each multiple produced by a prime has to percolate to the top eventually, so it's better to make its path shorter. It'll have to go through fewer merge nodes this way.

The linearity is imposed by type asymmetry of our (merge' :: a -> b -> b) function, forcing us into the 1+(1+(1+(1+...))) pattern, + standing for merge' (which was defined that way to prevent the run-ahead when folding over the infinite list of lists of multiples).

We need to turn it into an associative operation of uniform type (:: a -> a -> a) to be able to freely rearrange the combinations into arbitrary tree-like patterns, as in e.g. ((1+1)+(2+2))+(...) etc. The type uniformity is what makes compositionality possible.

The code in the "Implicit Heap" section below improves on that, and is essentially equivalent to using a treefold instead of a standard linear foldr:

Treefold Merged Multiples Removal

  primes :: [Integer]
  primes = 2:primes'
    primes' = [3,5] ++ drop 2 [3,5..] `minus` comps
    mults   = map (\p-> let q=p*p in ([q],tail [q,q+2*p..])) $ primes'
    comps   = fst $ tfold mergeSP (pairwise mergeSP mults)

  pairwise f (x:y:ys)  = f x y : pairwise f ys

  tfold f (a: ~(b: ~(c:xs)))
                       = (a `f` (b `f` c)) `f` tfold f (pairwise f xs)

  mergeSP (a,b) ~(c,d) = let (bc,b') = spMerge b c
                         in (a ++ bc, merge b' d)
    spMerge :: (Ord a) => [a] -> [a] -> ([a],[a]) 
    spMerge a@(x:xs) b@(y:ys) = case compare x y of
            LT ->  (x:c,d)  where (c,d) = spMerge xs b
            EQ ->  (x:c,d)  where (c,d) = spMerge xs ys
            GT ->  (y:c,d)  where (c,d) = spMerge a  ys
    spMerge a [] = ([] ,a)
    spMerge [] b = ([] ,b)

The fold used here creates a (2+(2+2))+( (4+(4+4)) + ( (8+(8+8)) + ... )) structure, better adjusted for primes multiples production than 1+(2+(4+(8+...))), used by the "Implicit Heap" code, giving it additional 10% speedup.

mergeSP is an associative operation, preserving of the invariant such that for a list of multiples [(a,b),(c,d), ...], it's always the case that last a < head b && last a < head c. These "split pairs" represent ordered list as a pair of its known and (currently) finite prefix, and the rest of it. Such pairs under mergeSP operation form a monoid, and if we were to declare a newtype SplitPair a = SP ([a],[a]) a Monoid instance, with mergeSP its mappend, the above code for comps would just become SP (comps,_) = mconcat mults.

This code exhibits approximately O()..O() local asymptotic behavior (tested interpreted, in GHCi, for 10,000 to 300,000 primes produced). When compared with Melissa O'Neill's PQ code from the ZIP package which was modified to work on odds only as well, it is 3.2x times faster, with used memory reported about 2.5x times smaller.

It can be further improved with the wheel optimization (as described below), giving it a further performance boost of about 30%-35%:

Treefold Merged Multiples, with Wheel

  primes :: [Integer]
  primes = 2:3:5:7:primes' 
    primes' = [11,13] ++ drop 2 (wheelSupply ()) `minus` comps
    mults   = map (\p-> let q=p*p in ([q],tail [q,q+2*p..])) $ primes'
    comps   = fst $ tfold mergeSP (pairwise mergeSP mults)

  wheel = 2:4:2:4:6:2:6:4:2:4:6:6:2:6:4:2:6:4:6:8:4:2:4:2:

  roll n (x:wh)  = n : roll (n+x) wh

  wheelSupply () = roll 11 wheel

This runs at about 20% faster than the Priority Queue-based code as present in Melissa O'Neill's ZIP package, with *exactly* the same local asymptotic behavior of about O()..O() (tested interpreted, in GHCi, for 10,000 to 300,000 primes produced), with used memory reported about the same.

More Filtering Sieves

The primes list creation with divisibility testing can be reformulated in a few more ways, using the list of primes as it is being built (a la "circular programming").

Odd numbers, by Trial Division

This is also good for generating a few 100,000s primes (when GHC-compiled as well):

  primes :: [Integer]
  primes = 2: 3: filter isPrime [5,7..]
    isPrime n = all (notDivs n) $ takeWhile (\p-> p*p <= n) (tail primes)
    notDivs n p = n `mod` p /= 0

Instead of relying on nested filters, it tests each odd number by an explicit list of all the needed prime factors. But for each number tested it re-fetches this list anew which will be the same for the increasing spans of numbers between the successive squares of primes.

Generated Spans, by Nested Filters

The other way to go instead of concentrating on the numbers supply, is to directly work on the successive spans between the primes squares.

This version is a bit faster still, creating 158,000 primes (again, GHC-compiled) in the same time as the postponed filters does 100,000 primes:

  primes :: [Integer]
  primes = 2: 3: sieve [] (tail primes) 5
    notDivsBy d n     = n `mod` d /= 0
    sieve ds (p:ps) x = foldr (filter . notDivsBy) [x, x+2..p*p-2] ds
                        ++ sieve (p:ds) ps (p*p+2)

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 filters, which it repeatedly recreates.

Generated Spans, by List of Primes

The list of primes needed to test each range of odds is actually just the prefix of the primes list itself, of known length, and need not be specifically generated at all. Combined with one-call testing by the explicit list of primes, and direct generation of odds between the successive primes squares, this leads to:

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

It produces about 222,000 primes in the same amount of time, 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 
      (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).

Multiples Removal on Generated Spans, or Sieve of Eratosthenes

The divisibility testing too should be considered a specification (as in "no multiples of p"), and not a code per se, because although testing composites is cheap (as most of them will have small factors, so the test is soon aborted), testing prime numbers is costly, and is to be avoided.

All the filtering versions thus far try to keep the primes among all numbers by testing each number in isolation. Instead, all the relevant primes' multiples can be removed from the corresponding segments of odds, and what's left after that will be just primes:

  primes :: [Integer]
  primes = 2: 3: sieve [] (tail primes) 5
    sieve fs (p:ps) x = [x,x+2..q-2] `minus` foldl merge [] mults
                        ++ sieve ((2*p,q+2*p):fs') ps (q+2)
      q                       = p*p
      (mults,fs')             = un_zip $ map mark fs
      mark (s,y)              = (h,(s,y')) 
                                  where (h,~(y':_)) = span(<q)[y,y+s..]

This modifies the preceding sieve to "mark" the odd composites in a given range (instead of testing their divisibility) by generating them - just as a person performing the original sieve of Eratosthenes would do, marking one by one the multiples of the relevant primes.

Compared with the previous version, interpreted under both GHCi and WinHugs, it runs faster, takes less memory, and has better asymptotic behavior, its performance approximately the same as in the Merged Multiples Removal sieve. The advantage in working with spans explicitly is that this code is easily amendable to using arrays for the composites marking and removal on each finite span.

Using Arrays

Generating Primes Spans with Arrays

The removal of multiples on each segment of odds can be done with arrays, instead of lists with "minus" and "merge":

  primes :: [Integer]
  primes = 2: 3: sieve [] (tail primes) 5  
    sieve fs (p:ps) x = [i | i<- [x,x+2..q-2], a!i] 
                          ++ sieve ((2*p,q):fs') ps (q+2)
      q           = p*p
      mults       = [ [x+s,x+2*s..q] | (s,x)<- fs ]
      fs'         = [ (s,last ms)    | ((s,_),ms)<- zip fs mults]
      a           = accumArray (\a b->False) True (x,q-2) 
                                [(i,()) | ms<- mults, i<- ms]

Apparently, arrays are very fast. This code runs faster than the Treefold Merged with Wheel version (itself 20% faster than Melissa O'Neill's PQ version), by 2.0x for 10,000, 2.25x for 100,000 and 2.4x times faster for 300,000 primes produced (testing it interpreted in GHCi), with about twice less memory usage reported.

Calculating Primes Upto a Given Value

  primesToN n = 2: [i | i<-[3,5..n], ar!i]
    ar = f 5 $ accumArray (\a b->False) True (3,n) [(i,())|i<-[9,15..n]]
    f p a | q>=n = a
          | True = if null x then a' else f (head x) a'
      where q = p*p
            a'= a//[(i,False)|i<-[q,q+2*p..n]]
            x = [i | i<-[p+2,p+4..n], a' !i]

Calculating Primes in a Given Range

  primesFromTo a b = (if a<3 then [2] else []) 
                     ++ [i | i<-[o,o+2..b], ar!i]
    o  = max (if even a then a+1 else a) 3
    r  = (1+).floor.sqrt.fromInteger $ b
    ar = accumArray (\a b->False) True (o,b) 
          [(i,()) | p <- [3,5..r]
                    , let q  = p*p 
                          s  = 2*p 
                          (n,x) = quotRem (o-q) s 
                          q' = if   o <= q  then q
                               else if x==0 then q+n*s
                               else              q+(n+1)*s
                    , i<- [q',q'+s..b] ]

Although testing by odds instead of by primes, the array generation is so fast that it is very much feasible and even preferable for quick generation of some short spans of relatively big primes.

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) = x `f` 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
    multiples p = vip [p*p,p*p+2*p..]

    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.

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'
    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
    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.

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
./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..])))
                    end = I.findMax is

-- primes - gives all primes up to n 
primes n = worker 2 (I.fromAscList [2..n])
                    worker x is 
                     | (x*x) > n = is
                     | otherwise = worker (findNext (x+1) is) (mark x is)

Testing Primality

Primality Test and Integer Factorization

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
     go n ps@(p:pt)
        | p*p > n        = [n]
        | n `rem` p == 0 = p : go (n `quot` p) ps
        | otherwise      = go n pt

Miller-Rabin Primality Test

find2km :: Integral a => a -> (a,a)
find2km n = f 0 n
        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)
        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
        f x n y
            | n == 1 = x `mul` y
            | r == 0 = f x2 q y
            | otherwise = f x2 q (x `mul` y)
                (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.