Euler's Totient Function, also known as Euler's phi function, is a mathematical function used to determine the number of positive integers less than a given number that are relatively prime to that number.
Euler's Totient Function is calculated by taking the product of the given number and 1 minus the reciprocal of each prime factor of that number. For example, if the given number is 10, the prime factors are 2 and 5. Therefore, the calculation would be: 10 x (1 - 1/2) x (1 - 1/5) = 4.
Euler's Totient Function is used in number theory and cryptography to find the number of possible keys in a public key cryptography system. It is also used in other mathematical applications, such as determining the order of a group.
Some key properties of Euler's Totient Function include: it is multiplicative, meaning that for any two coprime numbers, the function value of their product is the product of their individual function values; it follows a recursive formula; and it is even for all numbers greater than 2.
Euler's Totient Function is closely related to prime numbers, as it takes into account the number of prime factors a given number has. If a number is prime, its totient function value is simply one less than the number itself. Additionally, the totient function value of two prime numbers multiplied together is equal to the product of their individual totient function values.