Difference between revisions of "Fermat pseudoprime"
From Number
(Created page with '{{base-relative pseudoprimality property}} ==Definition== Suppose <math>n</math> is a composite natural number and <math>a</math> is relatively prime to <math>n</math>. <math>n...') |
|||
| (3 intermediate revisions by the same user not shown) | |||
| Line 1: | Line 1: | ||
{{base-relative pseudoprimality property}} | {{base-relative pseudoprimality property}} | ||
| − | + | {{nottobeconfusedwith|[[Fermat prime]]}} | |
==Definition== | ==Definition== | ||
| Line 9: | Line 9: | ||
In other words, <math>n</math> divides <math>a^{n-1} - 1</math>, or, the order of <math>a</math> mod <math>n</math> divides <math>n - 1</math>. | In other words, <math>n</math> divides <math>a^{n-1} - 1</math>, or, the order of <math>a</math> mod <math>n</math> divides <math>n - 1</math>. | ||
| + | ==Facts== | ||
| + | |||
| + | * [[Formula for number of bases to which a number is a Fermat pseudoprime]] | ||
==Relation with other properties== | ==Relation with other properties== | ||
| Line 19: | Line 22: | ||
===Property when applied to one or more choice of base=== | ===Property when applied to one or more choice of base=== | ||
| − | * [[ | + | * [[Carmichael number]] is a number that is a Fermat pseudoprime for every (relatively prime) base. |
* [[Poulet number]] is a Fermat pseudoprime to base <math>2</math> (in particular, it needs to be an odd number). | * [[Poulet number]] is a Fermat pseudoprime to base <math>2</math> (in particular, it needs to be an odd number). | ||
Latest revision as of 21:36, 3 January 2012
Template:Base-relative pseudoprimality property This is not to be confused with Fermat prime
Contents
Definition
Suppose 
is a composite natural number and 
is relatively prime to . is termed a Fermat pseudoprime relative to base if we have:
.
In other words, divides 
, or, the order of mod divides 
.
Facts
Relation with other properties
Stronger properties
- Strong pseudoprime to a given base.
- Euler pseudoprime to a given base.
- Euler-Jacobi pseudoprime to a given base.
Property when applied to one or more choice of base
- Carmichael number is a number that is a Fermat pseudoprime for every (relatively prime) base.
- Poulet number is a Fermat pseudoprime to base
(in particular, it needs to be an odd number).
