Difference between revisions of "Poulet number"

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===Initial examples===
 
===Initial examples===
  
The first few Poulet numbers are <math>341, 561, 645, 1105, 1387, 1729, 1905, 2047</math>.
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The first few Poulet numbers are:
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<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:
 
These include, for instance:
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! Statement !! Kind of numbers it says are Poulet numbers !! Smallest example !! Proof idea
 
! Statement !! Kind of numbers it says are Poulet numbers !! Smallest example !! Proof idea
 
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| [[Mersenne number for prime or Poulet implies prime or Poulet]] || 2047 (<math>M_{11}</math>)|| <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 || Use that <math>x - 1</math> divides <math>x^m - 1</math> as a polynomial with integer coefficients.
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| [[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.
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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>.
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| [[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)
 
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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>) || 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>.
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| [[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
 
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| [[Infinitude of Poulet numbers]] || N/A || N/A || Use the Mersenne number iteratively, after having found at least one Poulet number.
 
| [[Infinitude of Poulet numbers]] || N/A || N/A || Use the Mersenne number iteratively, after having found at least one Poulet number.

Latest revision as of 21:23, 15 January 2012

Template:Pseudoprimality property

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 OEIS

These include, for instance:

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