# 99 questions/Solutions/39

### From HaskellWiki

(*) 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 the primes out.

**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 :)]

**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 = if a' <= q then q else snapUp a' q step ]) where primes' = tail primesTME -- external unbounded list of primes a' = if a<=3 then 3 else (if even a then a+1 else a) 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 in if r==0 then v else v-r+step -- 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 a few 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] (6,038 reductions, 11,923 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).