Difference between revisions of "Euler totient"

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Euler's totient function is the function <br />
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Euler's totient function $\phi$ is defined for $n=1,2,3,\ldots$ so that $\phi(n)$ equals the number of positive integers less than or equal to $n$ that are [[relatively prime]] to $n$.  
<center>$\phi(n) =$ # of positive integers $\leq n$ that are relatively prime to $n$.</center>
 
  
 
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=Properties=
 
=Properties=
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<strong>Theorem:</strong> The function $\phi$ obeys the formula
 
$$\phi(n) = \displaystyle\sum_{d|n} \mu(d) \dfrac{n}{d},$$
 
where the notation $d | n$ indicates that $d$ is a [[divisor]] of $n$ and $\mu$ is the [[Möbius function]].
 
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<strong>Proof:</strong> █
 
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<strong>Theorem:</strong> The function $\phi$ obeys the formula
 
$$\phi(n) = n \displaystyle\prod_{p | n} \left( 1 - \dfrac{1}{p} \right),$$
 
where the notation $p | n$ indicates that $p$ is a prime that divides $n$.
 
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<strong>Proof:</strong>  █
 
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<strong>Theorem:</strong> The following formula holds:
 
$$\phi(n) = n\lim_{s \rightarrow 1} \zeta(s) \displaystyle\sum_{d | n} \mu(d)(e^{\frac{1}{d}})^{s-1},$$
 
where $\zeta$ is the [[Riemann zeta function]] and $\mu$ is the [[Möbius function]], $e$ is the [[e | base of the exponential]] and the notation $d|n$ indicates that $d$ is any [[divisor]] of $n$.
 
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<strong>Proof:</strong>  █
 
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=Videos=
 
=Videos=

Revision as of 01:08, 22 June 2016

Euler's totient function $\phi$ is defined for $n=1,2,3,\ldots$ so that $\phi(n)$ equals the number of positive integers less than or equal to $n$ that are relatively prime to $n$.

Properties

Videos

Euler's Totient Function: what it is and how it works
Euler's Totient Theorem: What is Euler's Totient Theorem and Why is it useful?
Euler's Totient Function | How To Find Totient Of A Number Using Euler's Product Formula
Euler's Totient Function
Euler's totient function
Prime Factorisation and Euler Totient Function Part 14
Application of Euler Totient Function Part 16
Möbius and Euler totient functions
Euler Totient Theorem, Fermat Little Theorems
Euler's Phi Function
03 Modern cryptography 08 Euler's totient function

References

Abramowitz&Stegun