Euler phi function

The Euler phi function, also known as Euler's totient function, is a number theory function that counts the positive integers up to a given integer n that are relatively prime to n. Two numbers are relatively prime if their greatest common divisor is 1. For example, for n = 10, the numbers that are relatively prime to 10 are 1, 3, 7, and 9. Therefore, the Euler phi function of 10, denoted as φ(10), is 4.12

The function is denoted by the Greek letter phi, φ(n). It has various properties, including being a multiplicative function, meaning that if a and b are relatively prime, then φ(ab) = φ(a)φ(b). This property is crucial for many applications of the phi function in number theory and cryptography.12

Leonhard Euler, a prominent mathematician, introduced this function. He used it in his work on number theory, including his generalization of Fermat's Little Theorem, which states that if a is an integer relatively prime to n, then (a^{\phi(n)} \equiv 1 \pmod{n}).12