Euler's Totient Theorem
Euler's Totient Theorem is a theorem closely related to his totient function.
Let be Euler's totient function. If is a positive integer, is the number of integers in the range which are relatively prime to . If is an integer and is a positive integer relatively prime to ,Then .
This theorem is credited to Leonhard Euler. It is a generalization of Fermat's Little Theorem, which specifies it when is prime. For this reason it is also known as Euler's generalization or the Fermat-Euler theorem.
Consider the set of numbers such that the elements of the set are the numbers relatively prime to . It will now be proved that this set is the same as the set where . All elements of are relatively prime to so if all elements of are distinct, then has the same elements as In other words, each element of is congruent to one of .This means that → → as desired. Note that dividing by is allowed since it is relatively prime to .