Euler Totient Function

In this post I will explain yet another simple number theoretic function, called the Euler Totient Function. This is a simple function which is used indirectly in many problems in programming contests. The function  is denoted as phi(n).  phi(n) basically gives the number of integers greater than 0 and lesser than n+1 which are co-prime to the integer n. In other words, it is the cardinality of the set

S = { a: 1<a<n , gcd(a,n) = 1 }. To calculate this function, we will use a straight forward brute force approach. With a simple analysis, we can show

that the running time of this algorithm is O(sqrt(n)).

- Start with an estimate of the totient function as n
- Iterate for i from 2 to sqrt(n)
- If i | n , we know that all factors of i less than equal
  to n will have gcd with n as atleast i. Hence, they can
 never be in the set S. Hence remove them from the result.
- While i|n, divide n by i.
- After the iteration of i from 2 to sqrt(n),
   if n is a number greater than 1, remove all
   its multiples less than n also from the result

The analysis of this code is similar to  the analysis for prime factorization. Essentially at the core of the algorithm, we are finding prime factors of n and removing all its multiples. Hence the numbers left are the ones that are co-prime to n.

Hence, both correctness and the running time analysis follows from the prime factorization algorithm.

Link to a java implementation.


Number Theoretic Algorithms – Factorization

In the following series of posts, I will write about some of the number theoretic algorithms that one encounters in most of competitive programming. These are fundamental algorithms which most computer scientist should know.

In this post I will post an algorithm and a corresponding JAVA code to perform prime factorization of a give number.

The algorithm to perform prime factorization is a very trivial algorithm and mostly encountered by everyone atleast once in their high school.

- Iterate for i from 2 to sqrt(n)
- While i|n, divide n by i. Report i as a prime factor of n
- After the iteration of i from 2 to sqrt(n),
   if n is a number greater than 1, report that also as a prime factor

The key observation is that, this algorithm runs in O(square root (n) ).

We are able to iterate only upto sqrt(n) since, we can claim one of the following:

1) n is a prime number


2) If n has a prime factor p greater than sqrt(n), then there exists a corresponding prime factor p’ lesser than or equal to sqrt(n), such that

p*p’ = n

If (1) doesn’t hold, it implies that n is a composite number. And Let us assume (2) doesnt hold.

This implies for a prime p such that p|n, there is no p’ <= sqrt(n) such that p’|n.

In particular n/p > sqrt(n) . And since p>sqrt(n)

=> 1/p < 1/sqrt(n)

=> n/p <n/sqrt(n) = sqrt(n).

This provides the contradiction. Hence the claim above proves the correctness of the algorithm.

Link to a JAVA implementation of the above factorization.