99 questions/Solutions/31
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(**) Determine whether a given integer number is prime.
Well, a natural number k is a prime number if it is larger than 1 and no natural number n >= 2 with n^2 <= k is a divisor of k. However, we don't actually need to check all natural numbers n <= sqrt k. We need only check the primes p <= sqrt k:
isPrime :: Integral a => a > Bool isPrime k = k > 1 && foldr (\p r > p*p > k  k `rem` p /= 0 && r) True primesTME
This uses
{# OPTIONS_GHC O2 fnocse #}  treemerging Eratosthenes sieve  producing infinite list of all prime numbers primesTME = 2 : gaps 3 (join [[p*p,p*p+2*p..]  p < primes']) where primes' = 3 : gaps 5 (join [[p*p,p*p+2*p..]  p < primes']) join ((x:xs):t) = x : union xs (join (pairs t)) pairs ((x:xs):ys:t) = (x : union xs ys) : pairs t gaps k xs@(x:t)  k==x = gaps (k+2) t  True = k : gaps (k+2) xs
union
function is readily available from
 duplicatesremoving 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
Here is another solution, intended to be extremely short while still being reasonably fast.
isPrime :: (Integral a) => a > Bool isPrime n  n < 4 = n > 1 isPrime n = all ((/=0).mod n) $ 2:3:[x + i  x < [6,12..s], i < [1,1]] where s = floor $ sqrt $ fromIntegral n
This one does not go as far as the previous, but it does observe the fact that you only need to check numbers of the form 6k +/ 1 up to the square root. And according to some quick tests (nothing extensive) this version can run a bit faster in some cases, but slower in others; depending on optimization settings and the size of the input.
There is a subtle bug in the version above. I'm new here (the wiki and the language) and don't know how corrections are best made (here, or on discussion?). Anyway, the above version will fail on 25, because the bound of s is incorrect. It is x+i that is bounded by the sqrt of the argument, not x. This version will work correctly:
isPrime n  n < 4 = n /= 1 isPrime n = all ((/=0) . mod n) $ takeWhile (<= m) candidates where candidates = (2:3:[x + i  x < [6,12..], i < [1,1]]) m = floor . sqrt $ fromIntegral n