# Difference between revisions of "Euler's Totient Theorem"

m |
m (grammar mistake corrected) |
||

Line 2: | Line 2: | ||

== Theorem == | == Theorem == | ||

− | Let <math>\phi(n)</math> be [[Euler's totient function]]. | + | Let <math>\phi(n)</math> be [[Euler's totient function]]. If <math>n</math> is a positive integer, <math>\phi{(n)}</math> is the number of integers in the range <math>\{1,2,3\cdots{,n}\}</math> which are relatively prime to <math>n</math>. If <math>{a}</math> is an integer and <math>m</math> is a positive integer [[relatively prime]] to <math>a</math>,Then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>. |

== Credit == | == Credit == |

## Revision as of 03:03, 29 December 2015

**Euler's Totient Theorem** is a theorem closely related to his totient function.

## Contents

## 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 that is prime. For this reason it is also known as Euler's generalization or the Fermat-Euler theorem.

## Proof

Consider the set of numbers {} (mod m) 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 {} (mod m) 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 (mod m) → (mod m) → (mod m) as desired. Note that dividing by is allowed since it is relatively prime to .