totientti

Totientti, also known as Euler's totient function or phi function, is a multiplicative function in number theory that counts the positive integers up to a given integer n that are relatively prime to n. Two integers are relatively prime if their greatest common divisor (GCD) is 1. For example, the totient of 10, denoted as phi(10) or totientti(10), is 4 because the numbers less than or equal to 10 that are relatively prime to 10 are 1, 3, 7, and 9.

The totient function has several important properties. If p is a prime number, then phi(p) = p - 1. If p is a prime number and k is a positive integer, then phi(p^k) = p^k - p^(k-1) = p^k(1 - 1/p). If gcd(m, n) = 1, then phi(mn) = phi(m)phi(n). This multiplicative property allows for the calculation of phi(n) for any composite number n by first finding its prime factorization. If n = p1^a1 * p2^a2 * ... * pk^ak is the prime factorization of n, then phi(n) = n * (1 - 1/p1) * (1 - 1/p2) * ... * (1 - 1/pk).

Euler's totient function is fundamental in number theory and has significant applications, particularly in cryptography. Euler's theorem states that if a and n are relatively prime positive integers, then a^phi(n) is congruent to 1 modulo n. This theorem is a generalization of Fermat's Little Theorem and is a cornerstone of public-key cryptography systems like RSA.