matqkks said:What is most motivating and tangible way of introducing this function? Does it in itself have any real life applications that have an impact. I can only think of a^phi(n)=1 (mod n) which is powerful result but is this function used elsewhere.
The Euler Totient or Phi Function, denoted as φ(n), is a mathematical function that counts the number of positive integers less than or equal to n that are relatively prime to n. In other words, it calculates the number of numbers that are coprime to n.
The Euler Totient or Phi Function is calculated using the formula φ(n) = n * (1-1/p1) * (1-1/p2) * ... * (1-1/pk), where n is the given number and p1, p2, ..., pk are the distinct prime factors of n.
The Euler Totient or Phi Function is used in many areas of mathematics, including number theory, cryptography, and group theory. It has important applications in the RSA encryption algorithm and is also used to determine the order of elements in a group.
Two numbers are relatively prime if they do not have any common factors other than 1. In other words, their greatest common divisor (GCD) is 1.
The Euler Totient or Phi Function is related to the GCD through the formula φ(n) = n * (1-1/p1) * (1-1/p2) * ... * (1-1/pk), where n is the given number and p1, p2, ..., pk are the distinct prime factors of n. This means that the GCD of two numbers n and m can be calculated using the formula GCD(n, m) = mn/φ(n)φ(m).