Poulet number
From Number
Template:Pseudoprimality property
Contents
Definition
A Poulet number or Sarrus number is an odd composite number 
such that:
.
In other words, divides 
. Equivalently, is a Fermat pseudoprime modulo 
.
Occurrence
Initial examples
The first few Poulet numbers are 
.
These include, for instance:
- The Mersenne number
, which is a Poulet number on account of the fact that Mersenne number for prime or Poulet implies prime or Poulet.
- Three Carmichael numbers -- these are numbers that are pseudoprime to every relatively prime base. These are
. 
is also known as the Hardy-Ramanujan number, and is the smallest number expressible as the sum of two cubes in two distinct ways.
Infinitude
Further information: Infinitude of Poulet numbers
There are infinitely many Poulet numbers. This can be proved in many ways. For instance, Mersenne number for prime or Poulet implies prime or Poulet. This shows that if we find one Poulet number, we can iterate the operation of taking the Mersenne number and obtain infinitely many Poulet numbers.
Facts
Relation with other properties
Stronger properties
- Absolute pseudoprime (at least, for odd numbers).
