Prime numbers
In mathematics, amongst the natural numbers greater than 1, a prime number (or a prime) is such that has no divisors other than itself (and 1).
Contents
- 1 Prime Number Resources
- 2 Definition
- 3 Sieve of Eratosthenes
- 4 Turner's sieve - Trial division
- 5 Euler's Sieve
- 6 Using Immutable Arrays
- 7 Using Mutable Arrays
- 8 Implicit Heap
- 9 Prime Wheels
- 10 Using IntSet for a traditional sieve
- 11 Prime Counting Functions
- 12 Testing Primality, and Integer Factorization
- 13 One-liners
- 14 External links
Prime Number Resources
- At Wikipedia:
- HackageDB packages:
- arithmoi: Various basic number theoretic functions; efficient array-based sieves, Montgomery curve factorization ...
- 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.
- Papers:
- O'Neill, Melissa E., "The Genuine Sieve of Eratosthenes", Journal of Functional Programming, Published online by Cambridge University Press 9 October 2008 doi:10.1017/S0956796808007004.
Definition
In mathematics, amongst the natural numbers greater than 1, a prime number (or a prime) is such that has no divisors other than itself (and 1). The smallest prime is thus 2. Non-prime numbers are known as composite, i.e. those representable as product of two natural numbers greater than 1.
To find out a prime's multiples we can either a. test each new candidate number for divisibility by that prime, giving rise to a kind of trial division algorithm; or b. we can directly generate the multiples of a prime p by counting up from it in increments of p, resulting in a variant of the sieve of Eratosthenes.
The set of prime numbers is thus
- P = { n ∈ N_{2} : (∀ m ∈ N_{2}) ( (m | n) ⇒ m = n) }
- = { n ∈ N_{2} : (∀ m ∈ N_{2}) ( m < n ⇒ ¬(m | n)) }
- = { n ∈ N_{2} : (∀ m ∈ N_{2}) ( m ⋅ m ≤ n ⇒ ¬(m | n)) }
- = N_{2} \ { n ⋅ m : n,m ∈ N_{2} }
- = N_{2} \ N_{2} ⋅ N_{2}
- = N_{2} \ ⋃ { { n ⋅ m : m ∈ N_{n} } : n ∈ N_{2} }
- = N_{2} \ ⋃ { { n ⋅ n, n ⋅ n+n, n ⋅ n+n+n, ... } : n ∈ N_{2} }
- = N_{2} \ ⋃ { { p ⋅ p, p ⋅ p+p, p ⋅ p+p+p, ... } : p ∈ P }
- ( = N_{2} \ ⋃ { { p ⋅ m : m ∈ N_{p} } : p ∈ P } = N_{2} \ P ⋅ N_{2} )
- where N_{k} = { n ∈ N : n ≥ k }
Thus starting with 2, for each newly found prime we can eliminate from the rest of the numbers all the multiples of this prime, giving us the next available number as next prime. This is known as sieving the natural numbers, so that in the end all the composites are eliminated and what we are left with are just primes. (This is what the last formula is describing, though seemingly impredicative, because it is self-referential. But because N_{2} is well-ordered with the order preserved under addition, the formula is well-defined.)
Prototypically, it is
primes = -- foldl (\\) [2..] [[p+p, p+p+p..] | p <- primes]
map head $ scanl (\\) [2..] [[p, p+p..] | p <- primes]
where
(\\) = Data.List.Ordered.minus
Having (a chain of) direct-access mutable arrays indeed enables easy marking of these multiples as is usually done in imperative languages; but to get an efficient list-based code we have to be smart about combining those streams of multiples of each prime - which gives us also the memory efficiency in generating the results incrementally, one by one.
Short exposition is here.
Sieve of Eratosthenes
Simplest, bounded, very inefficient formulation:
import Data.List (\\) -- (\\) is set-difference for unordered lists
primesTo m = sieve [2..m]
where
sieve (x:xs) = x : sieve (xs \\ [x,x+x..m])
sieve [] = []
-- or:
= ps
where
ps = map head $ takeWhile (not.null)
$ scanl (\\) [2..m] [[p, p+p..m] | p <- ps]
The (unbounded) sieve of Eratosthenes calculates primes as integers above 1 that are not multiples of primes, i.e. not composite — whereas composites are found as enumeration of multiples of each prime, generated by counting up from prime's square in constant increments equal to that prime (or twice that much, for odd primes). This is much more efficient and runs at about n^{1.2} empirical orders of growth (corresponding to n (log n)^{2} log log n complexity, more or less, in n primes produced):
import Data.List.Ordered (minus, union, unionAll)
primes = 2 : 3 : minus [5,7..] (unionAll [[p*p, p*p+2*p..] | p <- tail primes])
{- Using `under n = takeWhile (<= n)`, with ordered increasing lists,
`minus`, `union` and `unionAll` satisfy, for any `n` and `m`:
under n (minus a b) == nub . sort $ under n a \\ under n b
under n (union a b) == nub . sort $ under n a ++ under n b
under n . unionAll . take m == under n . foldl union [] . take m
under n . unionAll == nub . sort . concat
. takeWhile (not.null) . map (under n) -}
The definition is primed with 2 and 3 as initial primes, to avoid the vicious circle.
The "big union" unionAll
function could be defined as the folding of (\(x:xs) -> (x:) . union xs)
; or it could use a Bool
array as a sorting and duplicates-removing device. The processing naturally divides up into the segments between successive squares of primes.
Stepwise development follows (the fully developed version is here).
Initial definition
First of all, working with ordered increasing lists, the sieve of Eratosthenes can be genuinely represented by
-- genuine yet wasteful sieve of Eratosthenes
-- primes = eratos [2.. ] -- unbounded
primesTo m = eratos [2..m] -- bounded, up to m
where
eratos [] = []
eratos (p:xs) = p : eratos (xs `minus` [p, p+p..])
-- eratos (p:xs) = p : eratos (xs `minus` map (p*) [1..])
-- eulers (p:xs) = p : eulers (xs `minus` map (p*) (p:xs))
-- turner (p:xs) = p : turner [x | x <- xs, rem x p /= 0]
-- fix ( map head . scanl minus [2..] . map (\p-> [p, p+p..]) )
This should be regarded more like a specification, not a code. It runs at empirical orders of growth worse than quadratic in number of primes produced. But it has the core defining features of the classical formulation of S. of E. as a. being bounded, i.e. having a top limit value, and b. finding out the multiples of a prime directly, by counting up from it in constant increments, equal to that prime.
The canonical list-difference minus
and duplicates-removing union
functions (cf. Data.List.Ordered) are:
-- ordered lists, difference and union
minus (x:xs) (y:ys) = case (compare x y) of
LT -> x : minus xs (y:ys)
EQ -> minus xs ys
GT -> minus (x:xs) ys
minus xs _ = xs
union (x:xs) (y:ys) = case (compare x y) of
LT -> x : union xs (y:ys)
EQ -> x : union xs ys
GT -> y : union (x:xs) ys
union xs [] = xs
union [] ys = ys
The name merge ought to be reserved for duplicates-preserving merging operation of the merge sort.
Analysis
So for each newly found prime p the sieve discards its multiples, enumerating them by counting up in steps of p. There are thus multiples generated and eliminated for each prime, and multiples in total, with duplicates, by virtues of prime harmonic series.
If each multiple is dealt with in time, this will translate into RAM machine operations (since we consider addition as an atomic operation). Indeed mutable random-access arrays allow for that. But lists in Haskell are sequential-access, and complexity of minus(a,b)
for lists is instead of of the direct access destructive array update. The lower the complexity of each minus step, the better the overall complexity.
So on k-th step the argument list (p:xs)
that the eratos
function gets, starts with the (k+1)-th prime, and consists of all the numbers ≤ m coprime with all the primes ≤ p. According to the M. O'Neill's article (p.10) there are such numbers.
It looks like for our intents and purposes. Since the number of primes below m is by the prime number theorem (where is a prime counting function), there will be n multiples-removing steps in the algorithm; it means total complexity of at least , or in n primes produced - much much worse than the optimal .
From Squares
But we can start each elimination step at a prime's square, as its smaller multiples will have been already produced and discarded on previous steps, as multiples of smaller primes. This means we can stop early now, when the prime's square reaches the top value m, and thus cut the total number of steps to around . This does not in fact change the complexity of random-access code, but for lists it makes it , or in n primes produced, a dramatic speedup:
primesToQ m = eratos [2..m]
where
eratos [] = []
eratos (p:xs) = p : eratos (xs `minus` [p*p, p*p+p..m])
-- eratos (p:xs) = p : eratos (xs `minus` map (p*) [p..div m p])
-- eulers (p:xs) = p : eulers (xs `minus` map (p*) (under (div m p) (p:xs)))
-- turner (p:xs) = p : turner [x | x<-xs, x<p*p || rem x p /= 0]
Its empirical complexity is around . This simple optimization works here because this formulation is bounded (by an upper limit). To start late on a bounded sequence is to stop early (starting past end makes an empty sequence – see warning below^{ 1}), thus preventing the creation of all the superfluous multiples streams which start above the upper bound anyway (note that Turner's sieve is unaffected by this). This is acceptably slow now, striking a good balance between clarity, succinctness and efficiency.
^{1}Warning: this is predicated on a subtle point of minus xs [] = xs
definition being used, as it indeed should be. If the definition minus (x:xs) [] = x:minus xs []
is used, the problem is back and the complexity is bad again.
Guarded
This ought to be explicated (improving on clarity, though not on time complexity) as in the following, for which it is indeed a minor optimization whether to start from p or p*p - because it explicitly stops as soon as possible:
primesToG m = 2 : sieve [3,5..m]
where
sieve (p:xs)
| p*p > m = p : xs
| otherwise = p : sieve (xs `minus` [p*p, p*p+2*p..])
-- p : sieve (xs `minus` map (p*) [p,p+2..])
-- p : eulers (xs `minus` map (p*) (p:xs))
(here we also flatly ignore all evens above 2 a priori.) It is now clear that it can't be made unbounded just by abolishing the upper bound m, because the guard can not be simply omitted without changing the complexity back for the worst.
Accumulating Array
So while minus(a,b)
takes operations for random-access imperative arrays and about operations here for ordered increasing lists of numbers, when using Haskell's immutable array for a one could expect the array update time to be nevertheless closer to if destructive update was used implicitly by compiler for an array being passed along as an accumulating state parameter:
{-# OPTIONS_GHC -O2 #-}
import Data.Array.Unboxed
primesToA m = sieve 3 (array (3,m) [(i,odd i) | i<-[3..m]]
:: UArray Int Bool)
where
sieve p a
| p*p > m = 2 : [i | (i,True) <- assocs a]
| a!p = sieve (p+2) $ a // [(i,False) | i <- [p*p, p*p+2*p..m]]
| otherwise = sieve (p+2) a
Indeed for unboxed arrays (suggested by Daniel Fischer; with regular, boxed arrays it is very slow), the above code runs pretty fast, but with empirical complexity of O(n^{1.15..1.45}) in n primes produced (for producing from few hundred thousands to few millions primes, memory usage also slowly growing). If the update for each index were an O(1) operation, the empirical complexity would be seen as diminishing, as O(n^{1.15..1.05}), reflecting the true, linearithmic complexity.
We could use explicitly mutable monadic arrays (see below) to remedy this, but we can also think about it a little bit more on the functional side of things still.
Postponed
Going back to guarded Eratosthenes, first we notice that though it works with minimal number of prime multiples streams, it still starts working with each prematurely. Fixing this with explicit synchronization won't change complexity but will speed it up some more:
primesPE1 = 2 : sieve [3..] primesPE1
where
sieve xs (p:pt) | q <- p*p , (h,t) <- span (< q) xs =
h ++ sieve (t `minus` [q, q+p..]) pt
-- h ++ turner [x | x<-t, rem x p>0] pt
Inlining and fusing span
and (++)
we get:
primesPE = 2 : sieve [3..] [[p*p, p*p+p..] | p <- primesPE]
where
sieve (x:xs) t@((q:cs):r)
| x < q = x : sieve xs t
| otherwise = sieve (minus xs cs) r
Since the removal of a prime's multiples here starts at the right moment, and not just from the right place, the code could now finally be made unbounded. Because no multiples-removal process is started prematurely, there are no extraneous multiples streams, which were the reason for the original formulation's extreme inefficiency.
Segmented
With work done segment-wise between the successive squares of primes it becomes
primesSE = 2 : ops
where
ops = sieve 3 9 ops [] -- odd primes
sieve x q ~(p:pt) fs =
foldr (flip minus) [x,x+2..q-2] -- chain of subtractions
[[y+s, y+2*s..q] | (s,y) <- fs] -- OR,
-- [x,x+2..q-2] `minus` foldl union [] -- subtraction of merged
-- [[y+s, y+2*s..q] | (s,y) <- fs] -- lists
++ sieve (q+2) (head pt^2) pt
((2*p,q):[(s,q-rem (q-y) s) | (s,y) <- fs])
This "marks" the odd composites in a given range by generating them - just as a person performing the original sieve of Eratosthenes would do, counting one by one the multiples of the relevant primes. These composites are independently generated so some will be generated multiple times.
The advantage to working in spans explicitly is that this code is easily amendable to using arrays for the composites marking and removal on each finite span; and memory usage can be kept in check by using fixed sized segments.
Segmented Tree-merging
Rearranging the chain of subtractions into a subtraction of merged streams (see below) and using tree-like folding structure, further speeds up the code and significantly improves its asymptotic time behavior (down to about , space is leaking though):
primesSTE = 2 : ops
where
ops = sieve 3 9 ops [] -- odd primes
sieve x q ~(p:pt) fs =
([x,x+2..q-2] `minus` joinST [[y+s, y+2*s..q] | (s,y) <- fs])
++ sieve (q+2) (head pt^2) pt
((++ [(2*p,q)]) [(s,q-rem (q-y) s) | (s,y) <- fs])
joinST (xs:t) = (union xs . joinST . pairs) t
where
pairs (xs:ys:t) = union xs ys : pairs t
pairs t = t
joinST [] = []
Segmented merging via an array
The removal of composites is easy with arrays. Starting points can be calculated directly:
import Data.List (inits, tails)
import Data.Array.Unboxed
primesSAE = 2 : sieve 2 4 (tail primesSAE) (inits primesSAE)
-- (2:) . (sieve 2 4 . tail <*> inits) $ primesSAE
where
sieve r q ps (fs:ft) = [n | (n,True) <- assocs (
accumArray (\ _ _ -> False) True (r+1,q-1)
[(m,()) | p <- fs, let s = p * div (r+p) p,
m <- [s,s+p..q-1]] :: UArray Int Bool )]
++ sieve q (head ps^2) (tail ps) ft
The pattern of iterated calls to tail
is captured by a higher-order function tails
, which explicitly generates the stream of tails of a stream, making for a bit more readable (even if possibly a bit less efficient) code:
psSAGE = 2 : [n | (r:q:_, fs) <- (zip . tails . (2:) . map (^2) <*> inits) psSAGE,
(n,True) <- assocs (
accumArray (\_ _ -> False) True (r+1, q-1)
[(m,()) | p <- fs, let s = (r+p)`div`p*p,
m <- [s,s+p..q-1]] :: UArray Int Bool )]
Linear merging
But segmentation doesn't add anything substantially, and each multiples stream starts at its prime's square anyway. What does the Postponed code do, operationally? With each prime's square passed by, there emerges a nested linear left-deepening structure, (...((xs-a)-b)-...), where xs is the original odds-producing [3,5..] list, so that each odd it produces must go through more and more minus
nodes on its way up - and those odd numbers that eventually emerge on top are prime. Thinking a bit about it, wouldn't another, right-deepening structure, (xs-(a+(b+...))), be better? This idea is due to Richard Bird, seen in his code presented in M. O'Neill's article, equivalent to:
primesB = 2 : minus [3..] (foldr (\p r-> p*p : union [p*p+p, p*p+2*p..] r)
[] primesB)
or,
primesLME1 = 2 : prs
where
prs = 3 : minus [5,7..] (joinL [[p*p, p*p+2*p..] | p <- prs])
joinL ((x:xs):t) = x : union xs (joinL t)
Here, xs stays near the top, and more frequently odds-producing streams of multiples of smaller primes are above those of the bigger primes, that produce less frequently their multiples which have to pass through more union
nodes on their way up. Plus, no explicit synchronization is necessary anymore because the produced multiples of a prime start at its square anyway - just some care has to be taken to avoid a runaway access to the indefinitely-defined structure, defining joinL
(or foldr
's combining function) to produce part of its result before accessing the rest of its input (thus making it productive).
Melissa O'Neill introduced double primes feed to prevent unneeded memoization (a memory leak). We can even do multistage. Here's the code, faster still and with radically reduced memory consumption, with empirical orders of growth of around ~ (initially better, yet worsening for bigger ranges):
primesLME = 2 : _Y ((3:) . minus [5,7..] . joinL . map (\p-> [p*p, p*p+2*p..]))
_Y :: (t -> t) -> t
_Y g = g (_Y g) -- multistage, non-sharing, g (g (g (g ...)))
-- g (let x = g x in x) -- two g stages, sharing
_Y
is a non-sharing fixpoint combinator, here arranging for a recursive "telescoping" multistage primes production (a tower of producers).
This allows the primesLME
stream to be discarded immediately as it is being consumed by its consumer. For prs
from primesLME1
definition above it is impossible, as each produced element of prs
is needed later as input to the same prs
corecursive stream definition. So the prs
stream feeds in a loop into itself and is thus retained in memory, being consumed by self much slower than it is produced. With multistage production, each stage feeds into its consumer above it at the square of its current element which can be immediately discarded after it's been consumed. (3:)
jump-starts the whole thing.
Tree merging
Moreover, it can be changed into a tree structure. This idea is due to Dave Bayer and Heinrich Apfelmus:
primesTME = 2 : _Y ((3:) . gaps 5 . joinT . map (\p-> [p*p, p*p+2*p..]))
-- joinL ((x:xs):t) = x : union xs (joinL t)
joinT ((x:xs):t) = x : union xs (joinT (pairs t)) -- set union, ~=
where pairs (xs:ys:t) = union xs ys : pairs t -- nub.sort.concat
gaps k s@(x:xs) | k < x = k:gaps (k+2) s -- ~= [k,k+2..]\\s,
| True = gaps (k+2) xs -- when null(s\\[k,k+2..])
This code is pretty fast, running at speeds and empirical complexities comparable with the code from Melissa O'Neill's article (about in number of primes n produced).
As an aside, joinT
is equivalent to infinite tree-like folding foldi (\(x:xs) ys-> x:union xs ys) []
:
Data.List.Ordered.foldt
of the data-ordlist package builds the same structure, but in a lazier fashion, consuming its input at the slowest pace possible. Here this sophistication is not needed (evidently).
Tree merging with Wheel
Wheel factorization optimization can be further applied, and another tree structure can be used which is better adjusted for the primes multiples production (effecting about 5%-10% at the top of a total 2.5x speedup w.r.t. the above tree merging on odds only, for first few million primes):
primesTMWE = [2,3,5,7] ++ _Y ((11:) . tail . gapsW 11 wheel
. joinT . hitsW 11 wheel)
gapsW k (d:w) s@(c:cs) | k < c = k : gapsW (k+d) w s -- set difference
| otherwise = gapsW (k+d) w cs -- k==c
hitsW k (d:w) s@(p:ps) | k < p = hitsW (k+d) w s -- intersection
| otherwise = scanl (\c d->c+p*d) (p*p) (d:w)
: hitsW (k+d) w ps -- k==p
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:
4:8:6:4:6:2:4:6:2:6:6:4:2:4:6:2:6:4:2:4:2:10:2:10:wheel
-- cycle $ zipWith (-) =<< tail $ [i | i <- [11..11+_210], gcd i _210 == 1]
-- where _210 = product [2,3,5,7]
The hitsW
function is there to find the starting point for rolling the wheel for each prime, but this can be found directly:
primesW = [2,3,5,7] ++ _Y ( (11:) . tail . gapsW 11 wheel . joinT .
map (\p ->
map (p*) . dropWhile (< p) $
scanl (+) (p - rem (p-11) 210) wheel) )
Seems to run about 1.4x faster, too.
Above Limit - Offset Sieve
Another task is to produce primes above a given value:
{-# OPTIONS_GHC -O2 -fno-cse #-}
primesFromTMWE primes m = dropWhile (< m) [2,3,5,7,11]
++ gapsW a wh2 (compositesFrom a)
where
(a,wh2) = rollFrom (snapUp (max 3 m) 3 2)
(h,p2:t) = span (< z) $ drop 4 primes -- p < z => p*p<=a
z = ceiling $ sqrt $ fromIntegral a + 1 -- p2>=z => p2*p2>a
compositesFrom a = joinT (joinST [multsOf p a | p <- h ++ [p2]]
: [multsOf p (p*p) | p <- t] )
snapUp v o step = v + (mod (o-v) step) -- full steps from o
multsOf p from = scanl (\c d->c+p*d) (p*x) wh -- map (p*) $
where -- scanl (+) x wh
(x,wh) = rollFrom (snapUp from p (2*p) `div` p) -- , if p < from
wheelNums = scanl (+) 0 wheel
rollFrom n = go wheelNums wheel
where
m = (n-11) `mod` 210
go (x:xs) ws@(w:ws2) | x < m = go xs ws2
| True = (n+x-m, ws) -- (x >= m)
A certain preprocessing delay makes it worthwhile when producing more than just a few primes, otherwise it degenerates into simple trial division, which is then ought to be used directly:
primesFrom m = filter isPrime [m..]
Map-based
Runs ~1.7x slower than TME version, but with the same empirical time complexity, ~ (in n primes produced) and same very low (near constant) memory consumption:
import Data.List -- based on
import qualified Data.Map as M -- http://stackoverflow.com/a/1140100
primesMPE :: [Integer]
primesMPE = 2 : mkPrimes 3 M.empty prs 9 -- postponed sieve enlargement
where -- by decoupled primes feed loop
prs = 3 : mkPrimes 5 M.empty prs 9
mkPrimes n m ps@ ~(p:pt) q = case (M.null m, M.findMin m) of
{ (False, (n2, skips)) | n == n2 ->
mkPrimes (n+2) (addSkips n (M.deleteMin m) skips) ps q
; _ -> if n < q
then n : mkPrimes (n+2) m ps q
else mkPrimes (n+2) (addSkip n m (2*p)) pt (head pt^2)
}
addSkip n m s = M.alter (Just . maybe [s] (s:)) (n+s) m
addSkips = foldl' . addSkip
Turner's sieve - Trial division
David Turner's (SASL Language Manual, 1983) formulation replaces non-standard minus
in the sieve of Eratosthenes by stock list comprehension with rem
filtering, turning it into a trial division algorithm, for clarity and simplicity:
-- unbounded sieve, premature filters
primesT = sieve [2..]
where
sieve (p:xs) = p : sieve [x | x <- xs, rem x p > 0]
-- map head
-- $ iterate (\(p:xs) -> [x | x <- xs, rem x p > 0]) [2..]
This creates many superfluous implicit filters, because they are created prematurely. 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. To find e.g. the 1001st prime (7927
), 1000 filters are used, when in fact just the first 24 are needed (up to 89
's filter only). Operational overhead here is huge; theoretically, it has quadratic time complexity, in the number of produced primes.
Guarded Filters
But this really ought to be changed into the abortive variant, again achieving the "miraculous" complexity improvement from above quadratic to about empirically (in n primes produced) by stopping the sieving as soon as possible:
primesToGT m = sieve [2..m]
where
sieve (p:xs)
| p*p > m = p : xs
| True = p : sieve [x | x <- xs, rem x p > 0]
-- (\(a,b:_) -> map head a ++ b) . span ((< m).(^2).head) $
-- iterate (\(p:xs) -> [x | x <- xs, rem x p > 0]) [2..m]
Postponed Filters
Or it can remain unbounded, just filters creation must be postponed until the right moment:
primesPT1 = 2 : sieve primesPT1 [3..]
where
sieve (p:pt) xs = let (h,t) = span (< p*p) xs
in h ++ sieve pt [x | x <- t, rem x p > 0]
-- fix $ concatMap (fst . fst)
-- . iterate (\((_,xs), p:pt) -> let (h,t) = span (< p*p) xs in
-- ((h, [x | x <- t, rem x p > 0]), pt))
-- . (,) ([2],[3..])
It can be re-written with span
and (++)
inlined and fused into the sieve
:
primesPT = 2 : oddprimes
where
oddprimes = sieve [3,5..] 9 oddprimes
sieve (x:xs) q ps@ ~(p:pt)
| x < q = x : sieve xs q ps
| True = sieve [x | x <- xs, rem x p /= 0] (head pt^2) pt
creating here as well the linear filtering nested structure at run-time, (...(([3,5..] >>= filterBy [3]) >>= filterBy [5])...)
, but unlike the non-postponed code each filter being created at its proper moment, not sooner than the prime's square is seen.
filterBy ds n = [n | noDivs n ds] -- `ds` assumed to be non-decreasing
noDivs n ds = foldr (\d r -> d*d > n || (rem n d > 0 && r)) True ds
Optimal trial division
The above is algorithmically equivalent to the traditional formulation of trial division,
ps = 2 : [i | i <- [3..],
and [rem i p > 0 | p <- takeWhile (\p -> p^2 <= i) ps]]
or,
-- primes = filter (`noDivs`[2..]) [2..]
-- = 2 : filter (`noDivs`[3,5..]) [3,5..]
primesTD = 2 : 3 : filter (`noDivs` tail primesTD) [5,7..]
isPrime n = n > 1 && noDivs n primesTD
except that this code is rechecking for each candidate number which primes to use, whereas for every candidate number in each segment between the successive squares of primes these will just be the same prefix of the primes list being built.
Trial division is used as a simple primality test and prime factorization algorithm.
Segmented Generate and Test
Next we turn the list of filters into one filter by an explicit list, each one in a progression of prefixes of the primes list. This seems to eliminate most recalculations, explicitly filtering composites out from batches of odds between the consecutive squares of primes.
import Data.List
primesST = 2 : ops
where
ops = sieve 3 9 ops (inits ops) -- odd primes
-- (sieve 3 9 <*> inits) ops -- inits: [],[3],[3,5],...
sieve x q ~(_:pt) (fs:ft) =
filter ((`all` fs) . ((> 0).) . rem) [x,x+2..q-2]
++ sieve (q+2) (head pt^2) pt ft
This can also be coded as, arguably more readable,
primesSGT =
{- 2 : [ n | (px, r:q:_) <- zip (inits primesSGT)
(tails $ 2 : map (^2) primesSGT),
n <- [r+1..q-1], all ((> 0) . rem n) px ]
-}
2 : ops
where
ops = 3 : [n | (px, r:q:_) <- zip (inits ops) (tails $ 3 : map (^2) ops),
n <- [r+2,r+4..q-2], all ((> 0) . rem n) px]
-- n <- foldl (>>=) [r+2,r+4..q-2] -- chain of filters
-- [filterBy [p] | p <- px]] -- OR,
-- n <- [r+2,r+4..q-2] >>= filterBy px] -- a filter by a list
Generate and Test Above Limit
The following will start the segmented Turner sieve at the right place, using any primes list it's supplied with (e.g. TMWE etc.) or itself, as shown, demand computing it just up to the square root of any prime it'll produce:
primesFromST m | m <= 2 = 2 : primesFromST 3
primesFromST m | m > 2 =
sieve (m`div`2*2+1) (head ps^2) (tail ps) (inits ps)
where
(h,ps) = span (<= (floor.sqrt $ fromIntegral m+1)) ops
sieve x q ps (fs:ft) =
filter ((`all` (h ++ fs)) . ((> 0) .) . rem) [x,x+2..q-2]
++ sieve (q+2) (head ps^2) (tail ps) ft
ops = 3 : primesFromST 5 -- odd primes
-- ~> take 3 $ primesFromST 100000001234
-- [100000001237,100000001239,100000001249]
This is usually faster than testing candidate numbers for divisibility one by one which has to re-fetch anew the needed prime factors to test by, for each candidate. Faster is the offset sieve of Eratosthenes on odds, and yet faster the one w/ wheel optimization, on this page.
Conclusions
All these variants being variations of trial division, finding out primes by direct divisibility testing of every candidate number by sequential primes below its square root (instead of just by its factors, which is what direct generation of multiples is doing, essentially), are thus principally of worse complexity than that of Sieve of Eratosthenes.
The initial code is just a one-liner that ought to have been regarded as executable specification in the first place. It can easily be improved quite significantly with a simple use of bounded, guarded formulation to limit the number of filters it creates, or by postponement of filter creation.
Euler's Sieve
Unbounded Euler's sieve
With each found prime Euler's sieve removes all its multiples in advance so that at each step the list to process is guaranteed to have no multiples of any of the preceding primes in it (consists only of numbers coprime with all the preceding primes) and thus starts with the next prime:
primesEU = 2 : eulers [3,5..]
where
eulers (p:xs) = p : eulers (xs `minus` map (p*) (p:xs))
-- eratos (p:xs) = p : eratos (xs `minus` [p*p, p*p+2*p..])
This code is extremely inefficient, running above empirical complexity (and worsening rapidly), and should be regarded a specification only. Its memory usage is very high, with empirical space complexity just below , in n primes produced.
In the stream-based sieve of Eratosthenes we are able to skip along the input stream xs
directly to the prime's square, consuming the whole prefix at once, thus achieving the results equivalent to the postponement technique, because the generation of the prime's multiples is independent of the rest of the stream.
But here in the Euler's sieve it is dependent on all xs
and we're unable in principle to skip along it to the prime's square - because all xs
are needed for each prime's multiples generation. Thus efficient unbounded stream-based implementation seems to be impossible in principle, under the simple scheme of producing the multiples by multiplication.
Wheeled list representation
The situation can be somewhat improved using a different list representation, for generating lists not from a last element and an increment, but rather a last span and an increment, which entails a set of helpful equivalences:
{- fromElt (x,i) = x : fromElt (x + i,i)
=== iterate (+ i) x
[n..] === fromElt (n,1)
=== fromSpan ([n],1)
[n,n+2..] === fromElt (n,2)
=== fromSpan ([n,n+2],4) -}
fromSpan (xs,i) = xs ++ fromSpan (map (+ i) xs,i)
{- === concat $ iterate (map (+ i)) xs
fromSpan (p:xt,i) === p : fromSpan (xt ++ [p + i], i)
fromSpan (xs,i) `minus` fromSpan (ys,i)
=== fromSpan (xs `minus` ys, i)
map (p*) (fromSpan (xs,i))
=== fromSpan (map (p*) xs, p*i)
fromSpan (xs,i) === forall (p > 0).
fromSpan (concat $ take p $ iterate (map (+ i)) xs, p*i) -}
spanSpecs = iterate eulerStep ([2],1)
eulerStep (xs@(p:_), i) =
( (tail . concat . take p . iterate (map (+ i))) xs
`minus` map (p*) xs, p*i )
{- > mapM_ print $ take 4 spanSpecs
([2],1)
([3],2)
([5,7],6)
([7,11,13,17,19,23,29,31],30) -}
Generating a list from a span specification is like rolling a wheel as its pattern gets repeated over and over again. For each span specification w@((p:_),_)
produced by eulerStep
, the numbers in (fromSpan w)
up to are all primes too, so that
eulerPrimesTo m = if m > 1 then go ([2],1) else []
where
go w@((p:_), _)
| m < p*p = takeWhile (<= m) (fromSpan w)
| True = p : go (eulerStep w)
This runs at about complexity, for n
primes produced, and also suffers from a severe space leak problem (IOW its memory usage is also very high).
Using Immutable Arrays
Generating Segments of Primes
The sieve of Eratosthenes' removal of multiples on each segment of odds can be done by actually marking them in an array, instead of manipulating ordered lists, and can be further sped up more than twice by working with odds only:
import Data.Array.Unboxed
primesSA :: [Int]
primesSA = 2 : oddprimes ()
where
oddprimes = (3 :) . sieve 3 [] . oddprimes
sieve x fs (p:ps) = [i*2 + x | (i,True) <- assocs a]
++ sieve (p*p) ((p,0) :
[(s, rem (y-q) s) | (s,y) <- fs]) ps
where
q = (p*p-x)`div`2
a :: UArray Int Bool
a = accumArray (\ b c -> False) True (1,q-1)
[(i,()) | (s,y) <- fs, i <- [y+s, y+s+s..q]]
Runs significantly faster than TMWE and with better empirical complexity, of about in producing first few millions of primes, with constant memory footprint.
Calculating Primes Upto a Given Value
Equivalent to Accumulating Array above, running somewhat faster (compiled by GHC with optimizations turned on):
{-# OPTIONS_GHC -O2 #-}
import Data.Array.Unboxed
primesToNA n = 2: [i | i <- [3,5..n], ar ! i]
where
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 a2 else f (head x) a2
where q = p*p
a2 :: UArray Int Bool
a2 = a // [(i,False) | i <- [q, q+2*p..n]]
x = [i | i <- [p+2,p+4..n], a2 ! i]
Calculating Primes in a Given Range
primesFromToA a b = (if a<3 then [2] else [])
++ [i | i <- [o,o+2..b], ar ! i]
where
o = max (if even a then a+1 else a) 3 -- first odd in the segment
r = floor . sqrt $ fromIntegral b + 1
ar = accumArray (\_ _ -> False) True (o,b) -- initially all True,
[(i,()) | p <- [3,5..r]
, let q = p*p -- flip every multiple of an odd
s = 2*p -- to False
(n,x) = quotRem (o - q) s
q2 = if o <= q then q
else q + (n + signum x)*s
, i <- [q2,q2+s..b] ]
Although sieving 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.
Using Mutable Arrays
Using mutable arrays is the fastest but not the most memory efficient way to calculate prime numbers in Haskell.
Using ST Array
This method implements the Sieve of Eratosthenes, similar to how you might do it in C, modified to work on odds only. It is fast, but about linear in memory consumption, allocating one (though apparently packed) sieve array for the whole sequence to process.
import Control.Monad
import Control.Monad.ST
import Data.Array.ST
import Data.Array.Unboxed
sieveUA :: Int -> UArray Int Bool
sieveUA top = runSTUArray $ do
let m = (top-1) `div` 2
r = floor . sqrt $ fromIntegral top + 1
sieve <- newArray (1,m) True -- :: ST s (STUArray s Int Bool)
forM_ [1..r `div` 2] $ \i -> do
isPrime <- readArray sieve i
when isPrime $ do -- ((2*i+1)^2-1)`div`2 == 2*i*(i+1)
forM_ [2*i*(i+1), 2*i*(i+2)+1..m] $ \j -> do
writeArray sieve j False
return sieve
primesToUA :: Int -> [Int]
primesToUA top = 2 : [i*2+1 | (i,True) <- assocs $ sieveUA top]
Its empirical time complexity is improving with n (number of primes produced) from above towards . The reference C++ vector-based implementation exhibits this improvement in empirical time complexity too, from gradually towards , where tested (which might be interpreted as evidence towards the expected quasilinearithmic time complexity).
Using Page-Segmented ST-Mutable Unboxed Array
The following code is similar to the above in using (automatically by default) bit-packed unboxed arrays sieving "odds-only", but adds the feature of page-segmentation where each page is about the size of the CPU L2 cache (2^20 or about a million bits in the example) so can sieve reasonably efficiently without "cache thrashing" to this number squared or about 10^12. Using page segmentation, it is able to sieve to produce the primes to this limit only using the memory space required for three of these buffers, a (recursive) two used to sieve for the base primes up to a million (actually up to two million but only a million are used to this limit) and the third used for the page segments where they represent increasing values as the computation progresses.
Thus, only less than 400 Kilobytes is used sieving up to the limit of 10^12 rather than the about 62.5 Gigabytes required for the one-huge-monolithic array version (even bit-packed), and even if a computer had this much memory, the huge cost in memory access time of up up to about a hundred CPU clock cycles per access due to "cache-thrashing" as compared to the average of four or five CPU clock cycles as used in this code.
If one places the following GHC Haskell code in a file called TestPrimes.hs:
{-# OPTIONS_GHC -O2 #-}
import Data.Int ( Int64 )
import Data.Word ( Word64 )
import Data.Bits ( Bits(shiftR) )
import Data.Array.Base ( IArray(unsafeAt), UArray(UArray),
MArray(unsafeWrite), unsafeFreezeSTUArray )
import Control.Monad ( forM_ )
import Data.Array.ST ( MArray(newArray), runSTUArray )
type Prime = Word64
cSieveBufferLimit :: Int
cSieveBufferLimit = 2^17 * 8 - 1 -- CPU L2 cache in bits
primes :: () -> [Prime]
primes() = 2 : _Y (listPagePrms . pagesFrom 0) where
_Y g = g (_Y g) -- non-sharing multi-stage fixpoint combinator
listPagePrms pgs@(hdpg@(UArray lwi _ rng _) : tlpgs) =
let loop i | i >= fromIntegral rng = listPagePrms tlpgs
| unsafeAt hdpg i = loop (i + 1)
| otherwise = let ii = lwi + fromIntegral i in
case fromIntegral $ 3 + ii + ii of
p -> p `seq` p : loop (i + 1) in loop 0
makePg lwi bps = runSTUArray $ do
let limi = lwi + fromIntegral cSieveBufferLimit
bplmt = floor $ sqrt $ fromIntegral $ limi + limi + 3
strta bp = let si = fromIntegral $ (bp * bp - 3) `shiftR` 1
in if si >= lwi then fromIntegral $ si - lwi else
let r = fromIntegral (lwi - si) `mod` bp
in if r == 0 then 0 else fromIntegral $ bp - r
cmpsts <- newArray (lwi, limi) False
fcmpsts <- unsafeFreezeSTUArray cmpsts
let cbps = if lwi == 0 then listPagePrms [fcmpsts] else bps
forM_ (takeWhile (<= bplmt) cbps) $ \ bp ->
forM_ (let sp = fromIntegral $ strta bp
in [ sp, sp + fromIntegral bp .. cSieveBufferLimit ]) $ \c ->
unsafeWrite cmpsts c True
return cmpsts
pagesFrom lwi bps = map (`makePg` bps)
[ lwi, lwi + fromIntegral cSieveBufferLimit + 1 .. ]
main :: IO ()
main = print $ last $ take (10^ (10 :: Int)) $ primes()
Run as (for Linux-like systems):
$ ghc TestPrimes
$ time ./TestPrimes
179424673
real 0m0.714s
user 0m0.703s
sys 0m0.009s
The above code finds the first ten million primes in about 0.71 seconds, the first hundred million primes in about 7.6 seconds, the first thousand million (billion) primes in about 85.7, and the first ten billion primes in about 1087 seconds (Intel i5-6500 at 3.6 Ghz single threaded boost) for a empirical growth factor of about 1.1 to 1.2, which is about the theoretical limit.
Most of the above times are the time taken for the primes list processing, where the actual culling time is some fraction of the total time, with further optimizations plus multi-threading on multi-core CPU's able to make this small fraction even smaller. This is why conventional functional/incremental SoE's have very limited use for finding the first million primes or so as compared to use of mutable unboxed array based page-segmented SoE's which should be used for larger ranges, and have further optimizations reducing culling operation time by as much as an average of two times for these ranges, as well as the mentioned easy multi-threading by pages and further reductions in the number of operations through wheel-factorization, etc.
Implicit Heap
See Implicit Heap.
Prime Wheels
See Prime Wheels.
Using IntSet for a traditional sieve
See Using IntSet for a traditional sieve.
Prime Counting Functions
For counting of the number of primes up to a limit, just using a prime sieve is not the fastest way, especially when the ranges are over a few millions, and a prime counting function based on the inclusion-exclusion principle is much faster as although the operations are slower integer division operations rather than just simple adding and shifting integer operations, there are many less operations because the execution complexity is something like O(n^(2/3)) for Meissel based functions and O(n^(3/4)) for Legendre functions as per the one shown here. This operational complexity reduction comes about due to the inclusion-exclusion principle, which does not require knowledge of all the primes over the counting range as only a subset is required, which are found using a sieve. In this example, sieving to the square root of the counting range is a small part of the overall time so a basic odds-only Sieve of Eratosthenes is sufficient since most of the time will be spent determining the counts of primes to various points across the range.
The following code works by using the non-recursive partial sieving definition of the Legendre Phi function where for each base prime sieving pass up to the quad root of the counting range, a "smalls" Look Up Table (LUT) array of the counts of the remaining potential primes (which are only partially sieved, so may still be compound) is maintained, a LUT array of the remaining "roughs" or potential primes above the sieved base primes is updated, and a "larges" LUT array of the counting range divided by each of the remaining "roughs" is updated per sieving pass for each of the remaining potential "roughs". Following complete sieving, a final adjustment is made to compensate for the counting range divided by the product of pairs of primes above the quad root of the counting range where the primes must never be the same value. Since the LUT arrays represent actual count values (including the count of the sieving base primes) and not Phi values where the base primes are not included in the counts but one is, there is a compensation for this discrepancy at the end of each partial sieving loop and at the end of the "products of pairs" loop. Finally, after combining the counts into one result value, one is added to the count for the only even prime of two.
The following Legendre prime counting function code is iterative (not recursive although the iterative loops are expressed recursively) through the use of partial sieving where the culling array is sieved and counted one base prime at a time as is one interpretation of the Legendre Phi function (save as a file named PrimeCount.hs):
{-# OPTIONS_GHC -O2 -fllvm #-}
{-# LANGUAGE FlexibleContexts, BangPatterns #-}
import Data.Int (Int64, Int32)
import Data.Bits (Bits(shiftR), (.|.))
import Control.Monad (forM_)
import Control.Monad.ST (ST, runST)
import Data.Array.Base (STUArray, MArray(unsafeNewArray_), newArray,
unsafeAt, unsafeFreezeSTUArray, unsafeRead, unsafeWrite)
primeCount :: Int64 -> Int64
primeCount n =
if n < 9 then (if n < 2 then 0 else (n + 1) `div` 2) else
let
{-# INLINE divide #-}
divide :: Int64 -> Int64 -> Int
divide nm d = truncate $ (fromIntegral nm :: Double) / fromIntegral d
{-# INLINE half #-}
half :: Int -> Int
half x = (x - 1) `shiftR` 1
rtlmt = floor $ sqrt (fromIntegral n :: Double)
mxndx = (rtlmt - 1) `div` 2
(npc, ns, smalls, roughs, larges) = runST $ do
mss <- unsafeNewArray_ (0, mxndx) :: ST s (STUArray s Int Int32)
forM_ [ 0 .. mxndx ] $ \ i -> unsafeWrite mss i (fromIntegral i)
mrs <- unsafeNewArray_ (0, mxndx) :: ST s (STUArray s Int Int32)
forM_ [ 0 .. mxndx ] $ \ i -> unsafeWrite mrs i (fromIntegral i * 2 + 1)
mls <- unsafeNewArray_ (0, mxndx) :: ST s (STUArray s Int Int64)
forM_ [ 0 .. mxndx ] $ \ i ->
let d = fromIntegral (i + i + 1)
in unsafeWrite mls i (fromIntegral (divide n d - 1) `div` 2)
skips <- newArray (0, mxndx) False :: ST s (STUArray s Int Bool)
let loop i !pc !cs =
let sqri = (i + i) * (i + 1) in
if sqri > mxndx then do
fss <- unsafeFreezeSTUArray mss
frs <- unsafeFreezeSTUArray mrs
fls <- unsafeFreezeSTUArray mls
return (pc, cs + 1, fss, frs, fls)
else do
v <- unsafeRead skips i
if v then loop (i + 1) pc cs else do
unsafeWrite skips i True
let bp = i + i + 1
cull c = if c > mxndx then return () else do
unsafeWrite skips c True; cull (c + bp)
part k !s =
if k > cs then return (s - 1) else do
p <- unsafeRead mrs k
t <- unsafeRead skips ((fromIntegral p - 1) `shiftR` 1)
if t then part (k + 1) s else do
olv <- unsafeRead mls k
let d = fromIntegral p * fromIntegral bp
adjv <- -- return olv
if d <= fromIntegral rtlmt then do
sv <- unsafeRead mss (fromIntegral d `shiftR` 1)
unsafeRead mls (fromIntegral sv - pc)
else do
sv <- unsafeRead mss (half (divide n d))
return (fromIntegral sv)
unsafeWrite mls s (olv - adjv + fromIntegral pc)
unsafeWrite mrs s p; part (k + 1) (s + 1)
k0 = ((rtlmt `div` bp) - 1) .|. 1
adjc !j k =
if k < bp then return () else do
c <- unsafeRead mss (k `shiftR` 1)
let ac = c - fromIntegral pc
e = (k * bp) `shiftR` 1
adj m = if m < e then adjc m (k - 2)
else do ov <- unsafeRead mss m
unsafeWrite mss m (ov - ac)
adj (m - 1)
adj j
cull sqri; his <- part 0 0; adjc mxndx k0; loop (i + 1) (pc + 1) his
loop 1 0 mxndx
ans0 = unsafeAt larges 0 + fromIntegral ((ns + 2 * (npc - 1)) * (ns - 1) `div` 2)
accum j !ans =
if j >= ns then ans else
let q = fromIntegral (unsafeAt roughs j)
m = n `div` q
e = fromIntegral (unsafeAt smalls (half (fromIntegral (m `div` q)))) - npc
comp k !ac =
if k > e then ac else
let p = fromIntegral (unsafeAt roughs k)
ci = half (fromIntegral (divide m p))
in comp (k + 1) (ac + fromIntegral (unsafeAt smalls ci))
in if e <= j then ans else
accum (j + 1) (comp (j + 1) ans - fromIntegral ((e - j) * (npc + j - 1)))
in accum 1 (ans0 - sum [ unsafeAt larges i | i <- [ 1 .. ns - 1 ] ]) + 1
main :: IO ()
main = print $ primeCount (10^(11 :: Int))
Run as (Linux-like OS's):
ghc PrimeCount
time ./PrimeCount
4118054813
real 0m0.033s
user 0m0.029s
sys 0m0.004s
This time of about 29 milliseconds to count the primes to 10^11 and the ability to count the primes to 10^16 in about 73 seconds (Intel i5-6500 at 3.6 GHZ) shows the empirical growth of about 0.68, which is why it can count extended count ranges so fast where using a sieve to count the range to 10^16 with a computational complexity a little above one would take over a thousand hours single-threaded as here, even with the best of constant-factor wheel-factorization and loop-operation-time-reduction optimizations. The main problem with this algorithm is that it uses memory as the square root of the counting range at about 800 Megabytes for the range to 10^16.
Other prime counting function algorithms such as Lagarias, Miller, and Odlyzko's algorithm use memory proportional to the cube root of the counting range and have a slightly further reduced computational complexity while not significantly adding to the cost per operation as well as supporting easy multi-threading, but the code is somewhat long to show on this page of basic algorithms.
Prime counting function algorithms can also be modified to produce the sum of the primes up to a limit.
Testing Primality, and Integer Factorization
See Testing primality:
One-liners
See primes one-liners.
External links
- A collection of prime generators; the file "ONeillPrimes.hs" contains one of the fastest pure-Haskell prime generators; code by Melissa O'Neill.
- WARNING: Don't use the priority queue from older versions of that file for your projects: it's broken and works for primes only by a lucky chance. The revised version of the file fixes the bug, as pointed out by Eugene Kirpichov on February 24, 2009 on the haskell-cafe mailing list, and fixed by Bertram Felgenhauer.
- test entries for (some of) the above code variants.
- Speed/memory comparison table for sieve of Eratosthenes variants.
- Empirical orders of growth on Wikipedia.