Difference between revisions of "Carmichael function"
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=== Examples === | === Examples === | ||
{{incomplete|section}} | {{incomplete|section}} | ||
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+ | Evaluate <math>2009^{2009}</math> (mod <math>1000</math>). | ||
+ | [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1363764#1363764] | ||
== Second Definition == | == Second Definition == |
Revision as of 21:33, 3 January 2009
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 (mod
).
[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
.