Euler's totient function

In number theory, Euler's totient function (or Euler's phi function), denoted as φ(n) or ϕ(n), is an arithmetic function that counts the positive integers less than or equal to n that are relatively prime to n. (These integers are sometimes referred to as totatives of n.) Thus, if n is a positive integer, then φ(n) is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd(n, k) = 1.

Euler's totient function is a multiplicative function, meaning that if two numbers m and n are coprime, then φ(mn) = φ(m) φ(n).

For example, let n = 9. Then gcd(9, 3) = gcd(9, 6) = 3 and gcd(9, 9) = 9. The other six numbers in the range 1 ≤ k ≤ 9, that is 1, 2, 4, 5, 7 and 8 are relatively prime to 9. Therefore, φ(9) = 6. As another example, φ(1) = 1 since gcd(1, 1) = 1.

Euler's phi function is important mainly because it gives the order of the multiplicative group of integers modulo n (the group of units of the ring ℤ/nℤ). It also plays a key role in the definition of the RSA encryption system.