Totient Function

What is Euler's Totient Function?

Number theory is one of the most important topics in the field of Math and can be used to solve a variety of problems. Many times one might have come across problems that relate to the prime factorization of a number, to the divisors of a number, to the multiples of a number and so on.

Euler's Totient function is a function that is related to getting the number of numbers that are coprime to a certain number $$X$$ that are less than or equal to it. In short , for a certain number $$X$$ we need to find the count of all numbers $$Y$$ where $$ gcd(X,Y)=1 $$ and $$1 \le Y \le X$$.

A naive method to do so would be to Brute-Force the answer by checking the gcd of $$X$$ and every number less than or equal to $$X$$ and then incrementing the count whenever a $$GCD$$ of $$1$$ is obtained. However, this can be done in a much faster way using Euler's Totient Function.

According to Euler's product formula, the value of the Totient function is below the product over all prime factors of a number. This formula simply states that the value of the Totient function is the product after multiplying the number $$N$$ by the product of $$(1-(1/p))$$ for each prime factor of $$N$$.

So,
$$ \phi(n)= n\prod_{\substack{p \text{ prime } p \vert n}} \left( 1- \frac{1}{p}\right) $$

Algorithm steps:

  • Generate a list of primes.
  • While dealing with a certain $$N$$, check and store all the primes that perfectly divide $$N$$.
  • Now, it is just needed to use these primes and the above formula to get the result.

Implementation:

set<> primes;
static void mark(int num,int max,int[] arr)
{
    int i=2,elem;
    while((elem=(num*i))<=max)
    {
        arr[elem-1]=1;
        i++;
    }
}
GeneratePrimes()
{
    int arr[max_prime];
    for(int i=1;i<arr.length;i++)   
    {
        if(arr[i]==0)
        {
            list.add(i+1);
            mark(i+1,arr.length-1,arr);
        }
    }
}
main()
{
    GeneratePrimes();
    int N=nextInt();
    int ans=N;
    for(int k:set)
    {
        if(N%k==0)
        {
            ans*=(1-1/k);
        }
    }
    print(ans);
}

There are a few subtle observations that one can make about Euler's Totient Function.

  • The sum of all values of Totient Function of all divisors of $$N$$ is equal to $$N$$.
  • The value of Totient function for a certain prime $$P$$ will always be $$P-1$$ as the number $$P$$ will always have a $$GCD$$ of $$1$$ with all numbers less than or equal to it except itself.
  • For 2 number A and B, if $$GCD(A,B)==1$$ then $$Totient (A) \times Totient(B)$$ = $$Totient(A \cdot B)$$.
Contributed by: Anand Jaisingh
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