Difference between revisions of "Carmichael function"
(→Examples) |
(→Examples) |
||
Line 22: | Line 22: | ||
{{incomplete|section}} | {{incomplete|section}} | ||
− | 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] | ||
Revision as of 15:50, 29 March 2010
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
.