Difference between revisions of "Euler's Totient Theorem"

m (changed "relatively prime to to each other" to "relatively prime to m")
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==Proof==
 
==Proof==
  
Consider the set of numbers <math>A = </math>{<math>n_1, n_2, ... n_{\phi(m)} </math>} (mod m) such that the elements of the [[set]] are the numbers relatively [[prime]] to each other.  
+
Consider the set of numbers <math>A = </math>{<math>n_1, n_2, ... n_{\phi(m)} </math>} (mod m) such that the elements of the [[set]] are the numbers relatively [[prime]] to <math>m</math>.  
 
It will now be proved that this set is the same as the set  <math>B = </math>{<math>an_1, an_2, ... an_{\phi(m)} </math>} (mod m) where <math> (a, m) = 1</math>. All elements of <math>B</math> are relatively prime to <math>m</math> so if all elements of <math>B</math> are distinct, then <math>B</math> has the same elements as <math>A</math>. This means that <math> n_1 n_2 ... n_{\phi(m)} \equiv an_1 \cdot an_2 ... an_{\phi(m)}</math>(mod m) → <math>a^{\phi (m)} \cdot (n_1 n_2 ... n_{\phi(m)}) \equiv n_1 n_2 ... n_{\phi(m)}</math> (mod m) → <math>a^{\phi (m)} \equiv 1</math> (mod m) as desired.
 
It will now be proved that this set is the same as the set  <math>B = </math>{<math>an_1, an_2, ... an_{\phi(m)} </math>} (mod m) where <math> (a, m) = 1</math>. All elements of <math>B</math> are relatively prime to <math>m</math> so if all elements of <math>B</math> are distinct, then <math>B</math> has the same elements as <math>A</math>. This means that <math> n_1 n_2 ... n_{\phi(m)} \equiv an_1 \cdot an_2 ... an_{\phi(m)}</math>(mod m) → <math>a^{\phi (m)} \cdot (n_1 n_2 ... n_{\phi(m)}) \equiv n_1 n_2 ... n_{\phi(m)}</math> (mod m) → <math>a^{\phi (m)} \equiv 1</math> (mod m) as desired.
  

Revision as of 18:31, 1 February 2015

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

Theorem

Let $\phi(n)$ be Euler's totient function. If ${a}$ is an integer and $m$ is a positive integer relatively prime to $a$,in other words If $n$ is a positive integer, $\phi{(n)}$ is the number of integers in the range $\{1,2,3\cdots{,n}\}$ which are relatively prime to $n$.Then ${a}^{\phi (m)}\equiv 1 \pmod {m}$.

Credit

This theorem is credited to Leonhard Euler. It is a generalization of Fermat's Little Theorem, which specifies that ${m}$ is prime. For this reason it is also known as Euler's generalization or the Fermat-Euler theorem.

Proof

Consider the set of numbers $A =${$n_1, n_2, ... n_{\phi(m)}$} (mod m) such that the elements of the set are the numbers relatively prime to $m$. It will now be proved that this set is the same as the set $B =${$an_1, an_2, ... an_{\phi(m)}$} (mod m) where $(a, m) = 1$. All elements of $B$ are relatively prime to $m$ so if all elements of $B$ are distinct, then $B$ has the same elements as $A$. This means that $n_1 n_2 ... n_{\phi(m)} \equiv an_1 \cdot an_2 ... an_{\phi(m)}$(mod m) → $a^{\phi (m)} \cdot (n_1 n_2 ... n_{\phi(m)}) \equiv n_1 n_2 ... n_{\phi(m)}$ (mod m) → $a^{\phi (m)} \equiv 1$ (mod m) as desired.

See also

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