# Difference between revisions of "Poulet number"

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In other words, <math>n</math> divides <math>2^{n-1} - 1</math>. Equivalently, <math>n</math> is a [[defining ingredient::Fermat pseudoprime]] modulo <math>2</math>. | In other words, <math>n</math> divides <math>2^{n-1} - 1</math>. Equivalently, <math>n</math> is a [[defining ingredient::Fermat pseudoprime]] modulo <math>2</math>. | ||

+ | |||

+ | ==Occurrence== | ||

+ | |||

+ | ===Initial examples=== | ||

+ | |||

+ | The first few Poulet numbers are: | ||

+ | |||

+ | <section begin="list"/>[[341]], [[561]], [[645]], [[1105]], [[1387]], [[1729]], [[1905]], [[2047]], <toggledisplay>[[2465]], [[2701]], [[2821]], 3277, 4033, 4369, 4371, 4681, 5461, 6601, 7957, 8321, 8481, 8911, 10261, 10585, 11305, 12801, 13741, 13747, 13981, 14491, 15709, 15841, 16705, 18705, 18721, 19951, 23001, 23377, 25761, 29341</toggledisplay> [[Oeis:A001567|View list on OEIS]]<section end="list"/> | ||

+ | |||

+ | These include, for instance: | ||

+ | |||

+ | * The [[Mersenne number]] <math>M_{11} = 2047</math>, which is a Poulet number on account of the fact that [[Mersenne number for prime or Poulet implies prime or Poulet]]. | ||

+ | * Three [[Carmichael number]]s -- these are numbers that are pseudoprime to every relatively prime base. These are <math>561, 1105, 1729</math>. <math>1729</math> 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|[[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== | ||

+ | |||

+ | {| class="sortable" border="1" | ||

+ | ! Statement !! Kind of numbers it says are Poulet numbers !! Smallest example !! Proof idea | ||

+ | |- | ||

+ | | [[Mersenne number for prime or Poulet implies prime or Poulet]] || <math>M_n = 2^n - 1</math> where <math>n</math> itself is a Poulet number OR <math>M_n</math> where <math>n</math> is prime and <math>M_n</math> isn't || 2047 (<math>M_{11}</math>) factors as <math>23 * 89</math> || Use that <math>x - 1</math> divides <math>x^m - 1</math> as a polynomial with integer coefficients. | ||

+ | |- | ||

+ | | [[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 || 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== | ==Relation with other properties== | ||

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===Stronger properties=== | ===Stronger properties=== | ||

− | * [[Weaker than::Absolute pseudoprime]] (at least, | + | * [[Weaker than::Absolute pseudoprime]] (at least, for odd numbers). |

## Latest revision as of 21:23, 15 January 2012

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 | where itself is a Poulet number OR where is prime and isn't | 2047 () factors as | Use that divides as a polynomial with integer coefficients. |

Composite Fermat number implies Poulet number | where is not prime |
4294967297 () factors as | Use that divides as a polynomial with integer coefficients, and also that divides because . |

Square of Wieferich prime is Poulet number | where is a prime such that | 1194649 (square of 1093) | Use that divides 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 | where and both are primes | 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).