Difference between revisions of "Carmichael function"
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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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=== Examples === | === Examples === | ||
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== See also == | == See also == | ||
Revision as of 15:56, 6 January 2008
There are two different functions that are both called 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
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 
.
