Difference between revisions of "Euler's Totient Theorem"
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== Credit == | == Credit == | ||
− | This theorem is credited to [[Leonhard Euler]]. It is a generalization of [[Fermat's Little Theorem]], which specifies | + | This theorem is credited to [[Leonhard Euler]]. It is a generalization of [[Fermat's Little Theorem]], which specifies it when <math>{m}</math> is prime. For this reason it is also known as Euler's generalization or the Fermat-Euler theorem. |
==Proof== | ==Proof== |
Revision as of 09:56, 26 June 2020
Euler's Totient Theorem is a theorem closely related to his totient function.
Contents
[hide]Theorem
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 .
Credit
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.
Proof
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 .