# Check Primality of a number in Haskell

Statement Define a function isPrime which accepts a number n, and returns True if n is prime and False otherwise.
The easiest way to test the primality of a number is to test whether it has any factor. This is usally done by dividing the given number by the smallest prime number i.e. 2 first. If this divides the number, we declare that the given number is not prime, else we repeat the process with next prime number.
However this approach requires that we must have a table of prime numbers. It is not possible to use this approach if such table is not available. To overcome this difficuly, one can choose to simply divide the given number by integers starting with 2 till one finds a factor. It has been proven that for a number N, one need not test beyond the integer ⌊√N⌋ (Why?).
Let us produce a list1 of numbers from 2 to ⌊√N⌋. Function mkList takes an integer and returns a list of integers. Note that, in function mkList, we are passing n (with type Integer) to a function sqrt which only works on Reals. Since, sqrt is only defined for reals, it is customary to convert its argument to a real before passing it to function sqrt. We have used fromInteger for this purpose.
After creating a such a list, we need to test whether any element of this list divides the given number N. Function isAnyFactor is written to determine this. One can see that it uses another function isFactor which tests if two numbers are coprime 2 or not. If they are coprime then it return False, else it returns True. In function isFactor, we built a list of Boolean values using isFactor recursively. Now, it follows that if all elements of this list are False, then there is no number between 1 and √– N which divides the number N, or, the number is prime.
One surely can do some tricks. Such as by looking at the last digit, we can discard all even numbers and multiple of 5 etc.
[sourcecode language="text"]
<pre>
{- We need to make a list of integers from 2 to square root of N -}
mkList :: Integer -> [Integer]
mkList n = [2..k]
where k = toInteger (ceiling $sqrt$ fromIntegral n )
{- Check if a number divides another number. -}
isFactor :: Integer -> Integer -> Bool
isFactor m n | mod m n == 0 = True
| otherwise = False
{- This function returns array of Bool indicating if the element from
the list created by mkList is a factor of the given number or not. -}
isAnyFactor :: Integer -> [Integer] -> [Bool]
isAnyFactor _ [] = [False]
isAnyFactor z (x:xs) = isFactor z x : isAnyFactor z xs
{- Now test for primality. If a given number is prime, then all of the
elements in the list returned ny isAnyFactor must be False -}
isPrime :: Integer -> Bool
isPrime n = not $or$ isAnyFactor n (mkList n)
[/sourcecode]

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