Euler totient function
This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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The Euler totient function, also known as the Euler-phi function, on a natural number , denoted , is defined in the following equivalent ways:
- as the number of positive integers not greater than that are coprime to .
- as the number of generators of the cyclic group of order .
Every natural number is the sum of the Euler totient function on positive divisors
For any natural number , we have , where the summation runs over all positive divisors of .
Evaluation on prime powers
For any prime and positive integer , the value of Euler totient function on is given by the formula . This is an immediate application of the previous fact.
The Euler totient function is multiplicative, that is, if and are coprime, then . For full proof, see Euler totient function is multiplicative.
Knowing the prime factorization of , we can evaluate by repeated applications of the above facts.
However, knowing only the distinct prime factors of the number is sufficient, because the value of Euler totient function on it is then given by the formula , where here the product is evaluated over all distinct prime factors of .