Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer $n$ that are relatively prime to $n$ . It is written using the Greek letter phi as $φ (n)$ or $ϕ (n)$ , and may also be called Euler's phi function. It can be defined more formally as the number of integers $k$ in the range $1 \leq k \leq n$ for which the greatest common divisor $gcd(n, k)$ is equal to 1. The integers $k$ of this form are sometimes referred to as totatives of $n$ .

For example, the totatives of $n = 9$ are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, because $gcd(9, 3) = gcd(9, 6) = 3$ and $gcd(9, 9) = 9$ . Therefore, $φ (9) = 6$ . As another example, $φ (1) = 1$ since for $n = 1$ the only integer in the range from 1 to $n$ is 1 itself, and $gcd(1, 1) = 1$ .

Euler's totient function is a multiplicative function, meaning that if two numbers $m$ and $n$ are relatively prime, then $φ (mn) = φ (m) φ (n)$ . This function gives the order of the multiplicative group of integers modulo $n$ (the group of units of the ring $ℤ / n ℤ$ ). It also plays a key role in the definition of the RSA encryption system.

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