Difference between revisions of "Carmichael function"
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== First Definition == | == First Definition == | ||
| − | The Carmichael function <math>\lambda</math> is defined at <math>n</math> to be the smallest [[positive integer]] <math>\lambda(n)</math> such that <math>a^{\lambda(n)} \equiv 1\pmod {n}</math> for all positive [[integer]]s <math>a</math> [[relatively prime]] to <math>n</math>. The [[order]] of <math>a\pmod {n}</math> always divides <math>\lambda(n)</math>. | + | The Carmichael function <math>\lambda</math> is defined at <math>n</math> to be the smallest [[positive integer]] <math>\lambda(n)</math> such that <math>a^{\lambda(n)} \equiv 1\pmod {n}</math> for all positive [[integer]]s <math>a</math> [[relatively prime]] to <math>n</math>. The [[Order_(group theory)|order]] of <math>a\pmod {n}</math> always divides <math>\lambda(n)</math>. |
This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function. | This function is also known as the ''reduced totient function'' or the ''least universal exponent'' function. | ||
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\end{cases}</math></p></center> | \end{cases}</math></p></center> | ||
| − | + | == Examples == | |
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| − | Evaluate <math>2009^{2009} | + | Evaluate <math>2009^{2009}\pmod{1000}</math>. |
[http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1363764#1363764] | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1363764#1363764] | ||
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The second definition of the Carmichael function is the least common multiples of all the factors of <math>\phi(n)</math>. It is written as <math>\lambda'(n)</math>. However, in the case <math>8|n</math>, we take <math>2^{\alpha-2}</math> as a factor instead of <math>2^{\alpha-1}</math>. | The second definition of the Carmichael function is the least common multiples of all the factors of <math>\phi(n)</math>. It is written as <math>\lambda'(n)</math>. However, in the case <math>8|n</math>, we take <math>2^{\alpha-2}</math> as a factor instead of <math>2^{\alpha-1}</math>. | ||
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== See also == | == See also == | ||
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* [[Modular arithmetic]] | * [[Modular arithmetic]] | ||
* [[Euler's totient theorem]] | * [[Euler's totient theorem]] | ||
| + | * [[Carmichael numbers]] | ||
[[Category:Functions]] | [[Category:Functions]] | ||
[[Category:Number theory]] | [[Category:Number theory]] | ||
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| + | {{stub}} | ||
Latest revision as of 10:56, 1 August 2022
There are two different functions called the Carmichael function. Both are similar to Euler's totient function 
.
First Definition
The Carmichael function 
is defined at 
to be the smallest positive integer 
such that 
for all positive integers 
relatively prime to . The order of 
always divides .
This function is also known as the reduced totient function or the least universal exponent function.
Suppose 
. We have
Examples
Evaluate 
.
[1]
Second Definition
The second definition of the Carmichael function is the least common multiples of all the factors of 
. It is written as 
. However, in the case 
, we take 
as a factor instead of 
.
See also
This article is a stub. Help us out by expanding it.
