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 [[ | + | 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. | ||
Revision as of 14:54, 26 February 2020
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
This article is a stub. Help us out by expanding it.
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 
.
Examples
This article is a stub. Help us out by expanding it.
