# 99 questions/Solutions/31

(Difference between revisions)

(**) 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 -fno-cse #-}
-- tree-merging 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```

The tree-merging Eratosthenes sieve here seems to strike a good balance between efficiency and brevity. More at Prime numbers haskellwiki page. The semi-standard `union` function is readily available from
Data.List.Ordered
package, put here just for reference:

```-- 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```