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

For example, the totatives of ${\displaystyle n=9}$ 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, since ${\displaystyle \gcd(9,3)=\gcd(9,6)=3}$ and ${\displaystyle \gcd(9,9)=9}$. Therefore, ${\displaystyle \varphi (9)=6}$. As another example, ${\displaystyle \varphi (1)=1}$ since for ${\displaystyle n=1}$ the only integer in the range from 1 to ${\displaystyle n}$ is 1 itself, and ${\displaystyle \gcd(1,1)=1}$.

Euler's totient function is a multiplicative function, meaning that if two numbers ${\displaystyle m}$ and ${\displaystyle n}$ are relatively prime, then ${\displaystyle \varphi (mn)=\varphi (m)\varphi (n)}$. This function gives the order of the multiplicative group of integers modulo n (the group of units of the ring ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$). It is also used for defining the RSA encryption system.

Leonhard Euler introduced the function in 1763. However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter ${\displaystyle \pi }$ to denote it: he wrote ${\displaystyle \pi D}$ for "the multitude of numbers less than ${\displaystyle D}$, and which have no common divisor with it". This definition varies from the current definition for the totient function at ${\displaystyle D=1}$ but is otherwise the same. The now-standard notation ${\displaystyle \varphi (A)}$ comes from Gauss's 1801 treatise Disquisitiones Arithmeticae, although Gauss did not use parentheses around the argument and wrote ${\displaystyle \varphi A}$. Thus, it is often called Euler's phi function or simply the phi function.

In 1879, J. J. Sylvester coined the term totient for this function, so it is also referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's.

The cototient of ${\displaystyle n}$ is defined as ${\displaystyle n-\varphi (n)}$. It counts the number of positive integers less than or equal to ${\displaystyle n}$ that have at least one prime factor in common with ${\displaystyle n}$.

There are several formulae for computing ${\displaystyle \varphi (n)}$.

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