Eulerképlet

Eulerképlet, also known as Euler's totient function, is a fundamental arithmetic function denoted by φ(n), where n is a positive integer. It is named after Leonhard Euler, a Swiss mathematician who introduced the function in the 18th century.

The Eulerképlet φ(n) is defined as the number of positive integers less than or equal to n that are relatively prime to n. Two numbers are relatively prime if their greatest common divisor is 1. For example, φ(9) = 6 because the numbers 1, 2, 4, 5, 7, and 8 are relatively prime to 9.

The Eulerképlet has several important properties. For a prime number p, φ(p) = p - 1. For any integer n, φ(n) is the product of the φ values of its prime factors. Additionally, φ(n) is the number of integers from 1 to n that have a multiplicative inverse modulo n.

The Eulerképlet is used in various areas of mathematics, including number theory, cryptography, and combinatorics. In cryptography, it is used in the RSA algorithm, where the totient function is used to generate encryption keys. The Eulerképlet is also used in Euler's theorem, which states that if a and n are coprime, then a^φ(n) ≡ 1 (mod n).

The Eulerképlet is a crucial concept in number theory and has been extensively studied and generalized by mathematicians. It is a fundamental tool in understanding the structure of the integers and their properties.