Poulet number
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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:
341, 561, 645, 1105, 1387, 1729, 1905, 2047, [SHOW MORE] View list on OEISThese 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
Statement | Kind of numbers it says are Poulet numbers | Smallest example | Proof idea |
---|---|---|---|
Mersenne number for prime or Poulet implies prime or Poulet | ![]() ![]() ![]() ![]() ![]() |
2047 (![]() ![]() |
Use that ![]() ![]() |
Composite Fermat number implies Poulet number | ![]() ![]() |
4294967297 (![]() ![]() |
Use that ![]() ![]() ![]() ![]() ![]() |
Square of Wieferich prime is Poulet number | ![]() ![]() ![]() |
1194649 (square of 1093) | Use that ![]() ![]() |
Product of successive primes in Cunningham chain of the second kind satisfying congruence conditions is Poulet number | ![]() ![]() ![]() |
2701 (product of 37 and 73) | apply Fermat's little theorem, condition for 2 to be a quadratic residue |
Infinitude of Poulet numbers | N/A | N/A | Use the Mersenne number iteratively, after having found at least one Poulet number. |
Relation with other properties
Stronger properties
- Absolute pseudoprime (at least, for odd numbers).