Carmichael number: Difference between revisions
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==Definition== | ==Definition== | ||
A composite number <math>n > 1</math> is termed an ''' | A composite number <math>n > 1</math> is termed an '''Carmichael number''' or '''absolute pseudoprime''' if it satisfies the following condition: | ||
* The [[defining ingredient::Liouville-lambda function]] of <math>n</math> divides <math>n - 1</math>. | * The [[defining ingredient::Liouville-lambda function]] of <math>n</math> divides <math>n - 1</math>. | ||
* For any natural number <math>a</math> relatively prime to <math>n</math>, <math>n</math> divides <math>a^{n-1} - 1</math>. | * For any natural number <math>a</math> relatively prime to <math>n</math>, <math>n</math> divides <math>a^{n-1} - 1</math>. | ||
* <math>n</math> is a [[defining ingredient::Fermat pseudoprime]] to any base relatively prime to it. | * <math>n</math> is a [[defining ingredient::Fermat pseudoprime]] to any base relatively prime to it. | ||
==Facts== | |||
* [[There are infinitely many Carmichael numbers]] | |||
Revision as of 22:38, 6 April 2009
Template:Pseudoprimality property
Definition
A composite number is termed an Carmichael number or absolute pseudoprime if it satisfies the following condition:
- The Liouville-lambda function of divides .
- For any natural number relatively prime to , divides .
- is a Fermat pseudoprime to any base relatively prime to it.
