# Prime numbers miscellaneous

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− | <haskell> primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..n-1]]] | + | <haskell>primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..n-1]]] |

primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..min j (n`div`j)]]] | primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..min j (n`div`j)]]] | ||

## Revision as of 18:43, 18 August 2013

For a context to this, please see Prime numbers.

## Contents |

## 1 Implicit Heap

The following is an original implicit heap implementation for the sieve of
Eratosthenes, kept here for historical record. Also, it implements more sophisticated, lazier scheduling. The Prime_numbers#Tree merging with Wheel section simplifies it, removing the `People a`

structure altogether, and improves upon it by using a folding tree structure better adjusted for primes processing, and a wheel optimization.

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.

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 where 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.

## 2 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 6*k* + 1 or 6*k* + 5. 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

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]

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

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 Euler's Sieve, or the functional pearl titled Lazy wheel sieves and spirals of primes for more.

## 3 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)

*(doesn't look like it runs very efficiently)*.

## 4 One-liners

primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..n-1]]] primes = [n | n<-[2..], not $ elem n [j*k | j<-[2..n-1], k<-[2..min j (n`div`j)]]] primes = nubBy (((>1).).gcd) [2..] primes = [n | n<-[2..], all ((> 0).rem n) [2..n-1]] primes = 2 : [n | n<-[3,5..], all ((> 0).rem n) [3,5..floor.sqrt$fromIntegral n]] primes = 2 : [n | n<-[3..], all ((> 0).rem n) $ takeWhile ((<= n).(^2)) primes] primes = 2 : 3 : [n | n<-[5,7..], foldr (\p r-> p*p>n || (rem n p>0 && r)) True $ tail primes] primes = 2 : fix (\xs-> 3 : [n | n<-[5,7..], foldr (\p r-> p*p>n || (rem n p>0 && r)) True xs]) primes = map head $ iterate (\(x:xs)-> filter ((> 0).(`rem`x)) xs) [2..] primes = 2 : unfoldr (\(x:xs)-> Just(x, filter ((> 0).(`rem`x)) xs)) [3,5..] primesTo n = foldl (\r x-> r `minus` [x*x, x*x+2*x..]) (2:[3,5..n]) [3,5..floor.sqrt$fromIntegral n] primesTo n = 2 : foldr (\r z-> if (head r^2) <= n then head r : z else r) [] (fix $ \rs-> [3,5..n] : [t `minus` [p*p, p*p+2*p..] | (p:t)<- rs]) primes = 2 : concat (unfoldr (\(xs,p:ps)-> let (h,t)=span (< p*p) xs in Just (h, (filter ((> 0).(`rem`p)) t, ps))) ([3,5..],[3,5..])) primes = 2 : 3 : concat (unfoldr (\(xs,p:ps)-> let (h,t)=span (< p*p) xs in Just (h, (t `minus` [p*p, p*p+2*p..], ps))) ([5,7..],tail primes)) primes = concatMap snd $ iterate (\((n, p:t@(q:_)),_) -> ((n+1,t), [x | let lst = take n $ tail primes, x <- [p*p+2, p*p+4 .. q*q-2], all ((/= 0).rem x) $ lst])) ((1, tail primes), [2,3,5,7]) primes = concatMap snd $ iterate (\((n, p2:t@(q2:_)),_) -> ((n+1,t), minus [p2+2, p2+4 .. q2-2] $ foldi union [] [ [o, o+2*i .. q2-1] | i <- take n $ tail primes, let o=p2-rem (p2-i) (2*i)+2*i] )) ((1, map (^2) $ tail primes), [2,3,5,7]) primes = let { sieve (x:xs) = x : sieve [n | n <- xs, rem n x > 0] } in sieve [2..] primes = let { sieve xs (p:ps) = let (h,t)=span (< p*p) xs in h ++ sieve (filter ((> 0).(`rem`p)) t) ps } in 2 : 3 : sieve [5,7..] (tail primes) primes = let { sieve xs (p:ps) = let (h,t)=span (< p*p) xs in h ++ sieve (t `minus` [p*p, p*p+2*p..]) ps } in 2 : 3 : sieve [5,7..] (tail primes) primes = 2 : minus [3..] (foldr (\(x:xs)->(x:).union xs) [] $ map (\x->[x*x, x*x+x..]) primes) primes = 2 : minus [3,5..] (foldi (\(x:xs)->(x:).union xs) [] $ map (\x->[x*x, x*x+2*x..]) [3,5..]) primes = 2 : _Y ( (3:) . minus [5,7..] -- unbounded Sieve of Eratosthenes . foldi (\(x:xs) ys-> x:union xs ys) [] . map (\p->[p*p, p*p+2*p..]) ) _Y g = g (_Y g)

`foldi`

is an infinitely right-deepening tree folding function found here.