Poulet number: Difference between revisions

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Template:Pseudoprimality property

Definition

A Poulet number or Sarrus number is an odd composite number $n$ such that:

$2^{n - 1} \equiv 1 mod n$ .

In other words, $n$ divides $2^{n - 1} - 1$ . Equivalently, $n$ is a Fermat pseudoprime modulo $2$ .

Occurrence

Initial examples

The first few Poulet numbers are $341, 561, 645, 1105, 1387, 1729, 1905, 2047$ .

These include, for instance:

The Mersenne number $M_{11} = 2047$ , 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 $561, 1105, 1729$ . $1729$ 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	$M_{n} = 2^{n} - 1$ where $n$ itself is a Poulet number OR $M_{n}$ where $n$ is prime and $M_{n}$ isn't	2047 ( $M_{11}$ ) factors as $23 * 89$	Use that $x - 1$ divides $x^{m} - 1$ as a polynomial with integer coefficients.
Composite Fermat number implies Poulet number	$F_{n} = 2^{2^{n}} + 1$ where $F_{n}$ is not prime	4294967297 ( $F_{5}$ ) factors as $641 * 6700417$	Use that $x - 1$ divides $x^{m} - 1$ as a polynomial with integer coefficients, and also that $2^{n + 1}$ divides $2^{2^{n}}$ because $n + 1 \leq 2^{n}$ .
Square of Wieferich prime is Poulet number	$p^{2}$ where $p$ is a prime such that $2^{p - 1} \equiv 1 (\mod p^{2})$	1194649 (square of 1093)	Use that $x - 1$ divides $x^{m} - 1$ as a polynomial with integer coefficients twice (in base, then in exponent)
Product of successive primes in Cunningham chain of the second kind satisfying congruence conditions is Poulet number	$p (2 p - 1)$ where $p \equiv 1 (\mod 12)$ and both $p, 2 p - 1$ are primes	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).

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 |-
 | [[Composite Fermat number implies Poulet number]] || <math>F_n = 2^{2^n} + 1</math> where <math>F_n</math> is ''not'' prime || 4294967297 (<math>F_5</math>) factors as <math>641* 6700417</math> || Use that <math>x - 1</math> divides <math>x^m - 1</math> as a polynomial with integer coefficients, and also that <math>2^{n+1}</math> divides <math>2^{2^n}</math> because <math>n + 1 \le 2^n</math>.
+|-
+| [[Square of Wieferich prime is Poulet number]] || <math>p^2</math> where <math>p</math> is a prime such that <math>2^{p-1} \equiv 1 \pmod {p^2}</math> || 1194649 (square of 1093) || Use that <math>x - 1</math> divides <math>x^m - 1</math> as a polynomial with integer coefficients twice (in base, then in exponent)
+|-
+| [[Product of successive primes in Cunningham chain of the second kind satisfying congruence conditions is Poulet number]] || <math>p(2p - 1)</math> where <math>p \equiv 1 \pmod {12}</math> and both <math>p,2p-1</math> are primes || 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.