Carmichael number: Difference between revisions
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* 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. | ||
==Occurrence== | |||
===Initial examples=== | |||
<section begin="list"/>[[561]], [[1105]], [[1729]], [[2465]], <toggledisplay>2821, 6601, 8911, 10585, 15841, 29341, 41041, 46657, 52633, 62745, 63973, 75361, 101101, 115921, 126217, 162401, 172081, 188461, 252601, 278545, 294409, 314821, 334153, 340561, 399001, 410041, 449065, 488881, 512461</toggledisplay>[[Oeis:A002997|View list on OEIS]]<section end="list"/> | |||
==Facts== | ==Facts== | ||
* [[There are infinitely many Carmichael numbers]] | * [[There are infinitely many Carmichael numbers]] | ||
Revision as of 21:15, 2 January 2012
Template:Pseudoprimality property
Definition
A composite number is termed an Carmichael number or absolute pseudoprime if it satisfies the following condition:
- The universal exponent (also called the Carmichael function) of divides .
- For any natural number relatively prime to , divides .
- is a Fermat pseudoprime to any base relatively prime to it.
Occurrence
Initial examples
561, 1105, 1729, 2465, [SHOW MORE]