Prime factors in Haskell

This is a good-performanced and easy-to-understand implementation, in which isPrime and primes are defined recursively, and primes will be cached by default. primeFactors definition is just a proper use of primes, the result will contains continuous-duplicated numbers, this feature makes it easy to count the number of each factor via (map (head &&& length) . group) and it's easy to unique it via (map head . group) :

isPrime :: Int -> Bool
primes :: [Int]

isPrime n | n < 2 = False
isPrime n = all (\p -> n `mod` p /= 0) . takeWhile ((<= n) . (^ 2)) $ primes
primes = 2 : filter isPrime [3..]

primeFactors :: Int -> [Int]
primeFactors n = iter n primes where
    iter n (p:_) | n < p^2 = [n | n > 1]
    iter n ps@(p:ps') =
        let (d, r) = n `divMod` p
        in if r == 0 then p : iter d ps else iter n ps'

And the usage:

> import Data.List
> import Control.Arrow

> primeFactors 12312
[2,2,2,3,3,3,3,19]

> (map (head &&& length) . group) (primeFactors 12312)
[(2,3),(3,4),(19,1)]

> (map head . group) (primeFactors 12312)
[2,3,19]

Until the dividend m < 2,

1. take the first divisor n from primes.
2. repeat dividing m by n while divisible.
3. take the next divisor n from primes, and go to 2.

The list of all divisors actually used are prime factors of original m.

Code:

-- | prime factors
--
-- >>> factors 13
-- [13]
-- >>> factors 16
-- [2,2,2,2]
-- >>> factors 60
-- [2,2,3,5]
--
factors :: Int -> [Int]
factors m = f m (head primes) (tail primes) where
  f m n ns
    | m < 2 = []
    | m `mod` n == 0 = n : f (m `div` n) n ns
    | otherwise = f m (head ns) (tail ns)

-- | primes
--
-- >>> take 10 primes
-- [2,3,5,7,11,13,17,19,23,29]
--
primes :: [Int]
primes = f [2..] where f (p : ns) = p : f [n | n <- ns, n `mod` p /= 0]

Update:

This replacement code improves performance by avoiding unnecessary evaluations:

factors m = f m (head primes) (tail primes) where
  f m n ns
    | m < 2 = []
    | m < n ^ 2 = [m]   -- stop early
    | m `mod` n == 0 = n : f (m `div` n) n ns
    | otherwise = f m (head ns) (tail ns)

primes can also be sped up drastically, as mentioned in Will Ness's comment:

primes = 2 : filter (\n-> head (factors n) == n) [3,5..]

A simple approach to determine the prime factors of n is to

• search for the first divisor d in [2..n-1]
• if D exists: return d : primeFactors(div n d)
• otherwise return n (since n is prime)

Code:

prime_factors :: Int -> [Int]

prime_factors 1 = []
prime_factors n
  | factors == []  = [n]
  | otherwise = factors ++ prime_factors (n `div` (head factors))
  where factors = take 1 $ filter (\x -> (n `mod` x) == 0) [2 .. n-1]

This obviously could use a lot of optimization (search only from 2 to sqrt(N), cache the prime numbers found so far and compute the division only for these etc.)

UPDATE

A slightly modified version using case (as suggested by @user5402):

prime_factors n =
  case factors of
    [] -> [n]
    _  -> factors ++ prime_factors (n `div` (head factors))
  where factors = take 1 $ filter (\x -> (n `mod` x) == 0) [2 .. n-1]

Haskell allows you to create infinite lists, that are mutually recursive. Let's take an advantage of this.

First let's create a helper function that divides a number by another as much as possible. We'll need it, once we find a factor, to completely eliminate it from a number.

import Data.Maybe (mapMaybe)

-- Divide the first argument as many times as possible by the second one.
divFully :: Integer -> Integer -> Integer
divFully n q | n `mod` q == 0   = divFully (n `div` q) q
             | otherwise        = n

Next, assuming we have somewhere the list of all primes, we can easily find factors of a numbers by dividing it by all primes less than the square root of the number, and if the number is divisible, noting the prime number.

-- | A lazy infinite list of non-trivial factors of all numbers.
factors :: [(Integer, [Integer])]
factors = (1, []) : (2, [2]) : map (\n -> (n, divisors primes n)) [3..]
  where
    divisors :: [Integer] -> Integer -> [Integer]
    divisors _ 1          = []   -- no more divisors
    divisors (p:ps) n
        | p^2 > n       = [n]  -- no more divisors, `n` must be prime
        | n' < n        = p : divisors ps n'    -- divides
        | otherwise     = divisors ps n'        -- doesn't divide
      where
        n' = divFully n p

Conversely, when we have the list of all factors of numbers, it's easy to find primes: They are exactly those numbers, whose only prime factor is the number itself.

-- | A lazy infinite list of primes.
primes :: [Integer]
primes = mapMaybe isPrime factors
  where
    -- |  A number is prime if it's only prime factor is the number itself.
    isPrime (n, [p]) | n == p  = Just p
    isPrime _                  = Nothing

The trick is that we start the list of factors manually, and that to determine the list of prime factors of a number we only need primes less then its square root. Let's see what happens when we consume the list of factors a bit and we're trying to compute the list of factors of 3. We're consuming the list of primes, taking 2 (which can be computed from what we've given manually). We see that it doesn't divide 3 and that since it's greater than the square root of 3, there are no more possible divisors of 3. Therefore the list of factors for 3 is [3]. From this, we can compute that 3 is another prime. Etc.