# 99 questions/Solutions/39

### From HaskellWiki

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gaps a' (join [[x,x+step..b] | p <- takeWhile (<= z) primes' |
gaps a' (join [[x,x+step..b] | p <- takeWhile (<= z) primes' |
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, let q = p*p ; step = 2*p |
, let q = p*p ; step = 2*p |
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− | x = if a' <= q then q else snapUp a' q step ]) |
+ | x = snapUp (max a' q) q step ]) |

where |
where |
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primes' = tail primesTME -- external unbounded list of primes |
primes' = tail primesTME -- external unbounded list of primes |
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− | a' = if a<=3 then 3 else (if even a then a+1 else a) |
+ | a' = snapUp (max 3 a) 1 2 |

z = floor $ sqrt $ fromIntegral b + 1 |
z = floor $ sqrt $ fromIntegral b + 1 |
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join (xs:t) = union xs (join (pairs t)) |
join (xs:t) = union xs (join (pairs t)) |
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| True = k : gaps (k+2) xs |
| True = k : gaps (k+2) xs |
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gaps k [] = [k,k+2..b] |
gaps k [] = [k,k+2..b] |
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− | snapUp v origin step = let r = rem (v-origin) step |
+ | snapUp v origin step = let r = rem (v-origin) step -- rem OK if v>=origin |

− | in if r==0 then v else v-r+step |
+ | in if r==0 then v else v+(step-r) |

-- duplicates-removing union of two ordered increasing lists |
-- duplicates-removing union of two ordered increasing lists |
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union (x:xs) (y:ys) = case (compare x y) of |
union (x:xs) (y:ys) = case (compare x y) of |
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> primesR 10100 10200 -- Sol.3 |
> primesR 10100 10200 -- Sol.3 |
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[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] |
[10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] |
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− | (5,428 reductions, 11,310 cells) |
+ | (5,497 reductions, 11,382 cells) |

> takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME -- TME of Q.31 |
> takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME -- TME of Q.31 |

## Revision as of 10:51, 3 June 2011

(*) A list of prime numbers.

Given a range of integers by its lower and upper limit, construct a list of all prime numbers in that range.

**Solution 1:**

primesR :: Integral a => a -> a -> [a] primesR a b = filter isPrime [a..b]

If we are challenged to give all primes in the range between a and b we simply take all numbers from a up to b and filter all the primes through.

This is good for *very narrow ranges* as Q.31's `isPrime`

tests by *trial division* using (up to) a memoized primes list produced by sieve of Eratosthenes to which it refers internally. So it'll be slower, but immediate.

**Solution 2:**

primes :: Integral a => [a] primes = let sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ] in sieve [2..] primesR :: Integral a => a -> a -> [a] primesR a b = takeWhile (<= b) $ dropWhile (< a) primes

Another way to compute the claimed list is done by using the *Sieve of Eratosthenes*. The `primes`

function generates a list of all (!) prime numbers using this algorithm and `primesR`

filter the relevant range out. [But this way is very slow and I only presented it because I wanted to show how nicely the *Sieve of Eratosthenes* can be implemented in Haskell :)]

*this is of course a famous case of executable specification, with all the implied pitfalls of inefficiency when (ab)used as if it were an actual code*.

**Solution 3:**

Use the *proper* Sieve of Eratosthenes from e.g. 31st question's solution (instead of the above sieve of Turner), adjusted to start its multiples production from the given start point:

{-# OPTIONS_GHC -O2 -fno-cse #-} -- tree-merging Eratosthenes sieve, primesTME of haskellwiki/prime_numbers, -- adjusted to produce primes in a given range (inclusive) primesR a b | b<a || b<2 = [] | otherwise = (if a <= 2 then [2] else []) ++ gaps a' (join [[x,x+step..b] | p <- takeWhile (<= z) primes' , let q = p*p ; step = 2*p x = snapUp (max a' q) q step ]) where primes' = tail primesTME -- external unbounded list of primes a' = snapUp (max 3 a) 1 2 z = floor $ sqrt $ fromIntegral b + 1 join (xs:t) = union xs (join (pairs t)) join [] = [] pairs (xs:ys:t) = (union xs ys) : pairs t pairs t = t gaps k xs@(x:t) | k==x = gaps (k+2) t | True = k : gaps (k+2) xs gaps k [] = [k,k+2..b] snapUp v origin step = let r = rem (v-origin) step -- rem OK if v>=origin in if r==0 then v else v+(step-r) -- duplicates-removing union of two ordered increasing lists 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 a b = a ++ b

*(This turned out to be quite a project, with some quite subtle points.)* It should be much better then taking a slice of a full sequential list of primes, as it won't try to generate any primes between the *square root of b* and *a*. To wit,

> primesR 10100 10200 -- Sol.3 [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (5,497 reductions, 11,382 cells) > takeWhile (<= 10200) $ dropWhile (< 10100) $ primesTME -- TME of Q.31 [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (140,313 reductions, 381,058 cells) > takeWhile (<= 10200) $ dropWhile (< 10100) $ sieve [2..] -- Sol.2 where sieve (n:ns) = n:sieve [ m | m <- ns, m `mod` n /= 0 ] [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (54,893,566 reductions, 79,935,263 cells, 6 garbage collections) > filter isPrime [10100..10200] -- Sol.1 [10103,10111,10133,10139,10141,10151,10159,10163,10169,10177,10181,10193] (15,750 reductions, 29,292 cells) -- isPrime: Q.31

(testing with Hugs of Nov 2002).

This solution is faster but not immediate. It has a certain preprocessing stage but then goes on fast to produce the whole range. To illustrate, to produce the 49 primes in 1000-wide range above 120200300100 it takes about 18 seconds on my oldish notebook for the 1st version, with the first number produced almost immediately (~ 0.4 sec); but this version spews up all 49 primes in one go after just under 1 sec.

**Solution 4.**

For *very wide* ranges, specifically when , we're better off just using the primes sequence itself, without any post-processing:

primes :: Integral a => [a] primes = primesTME -- of Q.31 primesR :: Integral a => a -> a -> [a] primesR a b = takeWhile (<= b) . dropWhile (< a) $ primes