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Difference between revisions of "Euler's totient function"

m (→‎Formulas)
m (→‎Formulas)
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== Formulas ==
 
== Formulas ==
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To derive the formula, let us first define the [[prime factorization]] of <math> n </math> as <math> n =\prod_{i=1}^{n}p_i^{e_i} =p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n} </math> where the <math>p_i </math> are distinct [[prime number]]s.  Now, we can use a [[PIE]] argument to count the number of numbers less than or equal to  <math> n </math> that are relatively prime to it.
+
To derive the formula, let us first define the [[prime factorization]] of <math> n </math> as <math> n =\prod_{i=1}^{m}p_i^{e_i} =p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m} </math> where the <math>p_i </math> are distinct [[prime number]]s.  Now, we can use a [[PIE]] argument to count the number of numbers less than or equal to  <math> n </math> that are relatively prime to it.
  
−
First, let's count the complement of what we want (i.e. all the numbers less than <math> n </math> that share a common factor with it).  There are <math> p_1^{e_1-1}p_2^{e_2}\cdots p_n^{e_n} </math> numbers less than <math> n </math> that are divisible by <math> p_1 </math>.  If we do the same for each <math> p_k </math> and add these up, we get  
+
First, let's count the complement of what we want (i.e. all the numbers less than <math> n </math> that share a common factor with it).  There are <math> p_1^{e_1-1}p_2^{e_2}\cdots p_m^{e_m} </math> numbers less than <math> n </math> that are divisible by <math> p_1 </math>.  If we do the same for each <math> p_k </math> and add these up, we get  
  
−
<center><math> p_1^{e_1-1}p_2^{e_2}\cdots p_n{e_n} + p_1^{e_1}p_2^{e_2-1}\cdots p_n^{e_n} + \cdots + p_1^{e_1}p_2^{e_2}\cdots p_n^{e_n - 1}.</math></center>
+
<center><math> p_1^{e_1-1}p_2^{e_2}\cdots p_m{e_m} + p_1^{e_1}p_2^{e_2-1}\cdots p_m^{e_m} + \cdots + p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m - 1}.</math></center>
  
 
We can factor out, though:
 
We can factor out, though:
  
−
<center><math> p_1^{e_1-1}p_2^{e_2-1}\cdots p_n^{e_n-1}(p_1+p_2+\cdots + p_n).</math></center>
+
<center><math> p_1^{e_1-1}p_2^{e_2-1}\cdots p_m^{e_m-1}(p_1+p_2+\cdots + p_m).</math></center>
  
 
But we are obviously overcounting.  We then subtract out those divisible by two of the <math> p_k </math>.  We continue with this PIE argument to figure out that the number of elements in the complement of what we want is
 
But we are obviously overcounting.  We then subtract out those divisible by two of the <math> p_k </math>.  We continue with this PIE argument to figure out that the number of elements in the complement of what we want is
  
−
<center><math>p_1^{e_1-1}p_2^{e_2-1}\cdots p_n^{e_n-1}[(p_1+p_2+\cdots+p_n)-(p_1p_2+p_1p_3+\cdots+p_{n-1}p_n)+\cdots+(-1)^{n+1}(p_1p_2\cdots p_n)]</math></center>
+
<center><math>p_1^{e_1-1}p_2^{e_2-1}\cdots p_m^{e_m-1}[(p_1+p_2+\cdots+p_m)-(p_1p_2+p_1p_3+\cdots+p_{m-1}p_m)+\cdots+(-1)^{m+1}(p_1p_2\cdots p_m)]</math></center>
  
 
which we can factor further as
 
which we can factor further as
  
−
<center><math>p_1^{e_1-1}p_2^{e_2-1}\cdots p_n^{e_n-1}(p_1-1)(p_2-1)\cdots(p_n-1).</math></center>
+
<center><math>p_1^{e_1-1}p_2^{e_2-1}\cdots p_m^{e_m-1}(p_1-1)(p_2-1)\cdots(p_m-1).</math></center>
  
 
Making one small adjustment, we write this as
 
Making one small adjustment, we write this as
  
−
<center><math> \phi(n) = n\left(1-\frac 1{p_1}\right)\left(1-\frac 1{p_2}\right)\cdots\left(1-\frac 1{p_n}\right).</math></center>
+
<center><math> \phi(n) = n\left(1-\frac 1{p_1}\right)\left(1-\frac 1{p_2}\right)\cdots\left(1-\frac 1{p_m}\right).</math></center>
  
 
Given the general [[prime factorization]] of <math>{n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_m^{e_m}</math>, one can compute <math>\phi(n)</math> using the formula <cmath>\phi(n) = n\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_m}\right)</cmath>
 
Given the general [[prime factorization]] of <math>{n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_m^{e_m}</math>, one can compute <math>\phi(n)</math> using the formula <cmath>\phi(n) = n\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_m}\right)</cmath>

Revision as of 09:29, 8 November 2007

Euler's totient function $\phi(n)$ applied to a positive integer $n$ is defined to be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. $\phi(n)$ is read "phi of n."

Formulas

To derive the formula, let us first define the prime factorization of $n$ as $n =\prod_{i=1}^{m}p_i^{e_i} =p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m}$ where the $p_i$ are distinct prime numbers. Now, we can use a PIE argument to count the number of numbers less than or equal to $n$ that are relatively prime to it.

First, let's count the complement of what we want (i.e. all the numbers less than $n$ that share a common factor with it). There are $p_1^{e_1-1}p_2^{e_2}\cdots p_m^{e_m}$ numbers less than $n$ that are divisible by $p_1$. If we do the same for each $p_k$ and add these up, we get

$p_1^{e_1-1}p_2^{e_2}\cdots p_m{e_m} + p_1^{e_1}p_2^{e_2-1}\cdots p_m^{e_m} + \cdots + p_1^{e_1}p_2^{e_2}\cdots p_m^{e_m - 1}.$

We can factor out, though:

$p_1^{e_1-1}p_2^{e_2-1}\cdots p_m^{e_m-1}(p_1+p_2+\cdots + p_m).$

But we are obviously overcounting. We then subtract out those divisible by two of the $p_k$. We continue with this PIE argument to figure out that the number of elements in the complement of what we want is

$p_1^{e_1-1}p_2^{e_2-1}\cdots p_m^{e_m-1}[(p_1+p_2+\cdots+p_m)-(p_1p_2+p_1p_3+\cdots+p_{m-1}p_m)+\cdots+(-1)^{m+1}(p_1p_2\cdots p_m)]$

which we can factor further as

$p_1^{e_1-1}p_2^{e_2-1}\cdots p_m^{e_m-1}(p_1-1)(p_2-1)\cdots(p_m-1).$

Making one small adjustment, we write this as

$\phi(n) = n\left(1-\frac 1{p_1}\right)\left(1-\frac 1{p_2}\right)\cdots\left(1-\frac 1{p_m}\right).$

Given the general prime factorization of ${n} = {p}_1^{e_1}{p}_2^{e_2} \cdots {p}_m^{e_m}$, one can compute $\phi(n)$ using the formula \[\phi(n) = n\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right) \cdots \left(1-\frac{1}{p_m}\right)\]

Identities

For prime p, $\phi(p)=p-1$, because all numbers less than ${p}$ are relatively prime to it.

For relatively prime ${a}, {b}$, $\phi{(a)}\phi{(b)} = \phi{(ab)}$.

In fact, we also have for any ${a}, {b}$ that $\phi{(a)}\phi{(b)}\gcd(a,b)=\phi{(ab)}\phi({\gcd(a,b)})$.

For any $n$, we have $\sum_{d|n}\phi(d)=n$ where the sum is taken over all divisors d of $n$.

See also

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