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99 questions/Solutions/34

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(**) Calculate Euler's totient function phi(m).

Euler's so-called totient function phi(m) is defined as the number of positive integers r (1 <= r < m) that are coprime to m.

totient 1 = 1
totient a = length $ filter (coprime a) [1..a-1]
 where coprime a b = gcd a b == 1

We take coprime from the previous exercise and give it to filter, which applies it to each element of a list from 1 to one less than the number, returning only those that are true. length tells us how many elements are in the resulting list, and thus how many elements are coprime to n

Or slightly more concise, using list comprehension:

totient n = length [x | x <- [1..n], coprime x n]

For very large numbers, however, the above algorithms become very slow. The Wikipedia article for Euler's totient function provides an algorithm that uses the number's prime factors.

import Data.List (nub)
import Data.Ratio
totient :: (Integral a) => a -> a
totient 1 = 1
totient n = numerator ratio `div` denominator ratio
 where ratio = foldl (\acc x -> acc * (1 - (1 % x))) 
                 (n % 1) $ nub (primeFactors n)

This example uses Data.Ratio to ensure no precision is lost. It also relies on a function primeFactors (not shown) that returns a list of all a number's prime factors.