The number of positive integers less than or equal to n that are relatively prime to n.
Divisors and Divisibility: Understanding the concepts of divisors and divisibility is essential when learning about the Euler's totient function. Divisors are the numbers that divide a given number without leaving a remainder, while divisibility refers to whether a number can be divided by another number without leaving a remainder.
Prime Numbers: Prime numbers are those numbers that are only divisible by 1 and themselves. These numbers play a crucial role in the Euler's totient function, as they help identify the prime factors of a number.
Factorization: Factorization is the process of breaking down a number into its prime factors. It is an essential foundation for understanding the Euler's totient function, as it helps identify the prime factors of a given number.
Multiplicative Functions: Multiplicative functions are those functions that have the property that f(ab) = f(a)f(b) for all positive integers a,b. The Euler's totient function is a multiplicative function.
The Sieve of Eratosthenes: The Sieve of Eratosthenes is a simple algorithm used to find all the prime numbers less than or equal to a given number. It is a useful tool for identifying the prime factors of a given number.
Euler's Theorem: Euler's theorem is a fundamental theorem in number theory that establishes a relationship between the Euler's totient function and modular arithmetic. It states that if a and n are coprime, then a^φ(n) ≡ 1 mod n, where φ(n) is the Euler's totient function.
Euler's Totient Function: The Euler's totient function is a function that counts the number of positive integers less than or equal to n that are coprime to n. It is denoted by φ(n) and is defined as φ(n) = {k|1≤k≤n, gcd(k,n)=1}.
Carmichael Function: The Carmichael function is a function that is similar to the Euler's totient function but is more general. It counts the number of positive integers less than or equal to n that are coprime to n, where n is an odd composite number.
RSA Algorithm: The RSA algorithm is a widely used encryption algorithm that relies on the properties of the Euler's totient function. It is commonly used in secure communication and electronic transactions.
Applications of Euler's Totient Function: The Euler's totient function has a wide range of applications in various fields, including cryptography, number theory, and computer science. Some examples include the RSA algorithm, Diffie-Hellman key exchange, and the Miller-Rabin primality test.