Poulet number - Revision history

Vipul: /* Initial examples */

2012-01-15T21:23:09Z

Initial examples

← Older revision		Revision as of 21:23, 15 January 2012
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	The first few Poulet numbers are:		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"/>		<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:

Vipul: /* Initial examples */

2012-01-02T21:17:18Z

Initial examples

← Older revision		Revision as of 21:17, 2 January 2012
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	The first few Poulet numbers are:		The first few Poulet numbers are:

	<section begin="list"/>[[341]], [[561]], [[645]], [[1105]], [[1387]], [[1729]], [[1905]], [[2047]]<section end="list"/>		<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:

Vipul: /* Occurrence */

2012-01-02T21:11:49Z

Occurrence

← Older revision		Revision as of 21:11, 2 January 2012
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	===Initial examples===		===Initial examples===

	The first few Poulet numbers are <~~math~~>341, 561, 645, 1105, 1387, 1729, 1905, 2047</~~math~~>.		The first few Poulet numbers are:

			<section begin="list"/>[[341]], [[561]], [[645]], [[1105]], [[1387]], [[1729]], [[1905]], [[2047]]<section end="list"/>

	These include, for instance:		These include, for instance:

Vipul: /* Facts */

2012-01-02T19:59:51Z

Facts

← Older revision		Revision as of 19:59, 2 January 2012
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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)		\| [[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		\| [[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.		\| [[Infinitude of Poulet numbers]] \|\| N/A \|\| N/A \|\| Use the Mersenne number iteratively, after having found at least one Poulet number.

Vipul: /* Facts */

2012-01-02T19:59:22Z

Facts

← Older revision		Revision as of 19:59, 2 January 2012
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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>.		\| [[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.		\| [[Infinitude of Poulet numbers]] \|\| N/A \|\| N/A \|\| Use the Mersenne number iteratively, after having found at least one Poulet number.

Vipul: /* Facts */

2012-01-02T19:08:35Z

Facts

← Older revision		Revision as of 19:08, 2 January 2012
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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
	\|-		\|-
	\| [[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.		\| [[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>) \|\| 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>.		\| [[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>.
	\|-		\|-
	\| [[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.

Vipul: /* Facts */

2012-01-02T19:07:33Z

Facts

← Older revision		Revision as of 19:07, 2 January 2012
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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.		\| [[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.
	\|-		\|-
	\| [[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>.		\| [[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>.
	\|-		\|-
	\| [[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.

Vipul: /* Facts */

2012-01-02T19:07:15Z

Facts

← Older revision		Revision as of 19:07, 2 January 2012
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Statement !! Kind of numbers it says are Poulet numbers !! Proof idea		! 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 \|\| Use that <math>x - 1</math> divides <math>x^m - 1</math> as a polynomial with integer coefficients.		\| [[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.
	\|-		\|-
	\| [[Composite Fermat number implies Poulet number]] \|\| <math>F_n = 2^{2^n} + 1</math> where <math>F_n</math> is ''not'' prime \|\| 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>.		\| [[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>.
	\|-		\|-
	\| [[Infinitude of Poulet numbers]] \|\| \|\| 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.
	\|}		\|}

Vipul: /* Facts */

2012-01-02T19:05:16Z

Facts

← Older revision		Revision as of 19:05, 2 January 2012
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	==Facts==		==Facts==

	* [[Mersenne number for prime or Poulet implies prime or Poulet]]		{\| class="sortable" border="1"
	* [[Composite Fermat number implies Poulet number]]		! Statement !! Kind of numbers it says are Poulet numbers !! Proof idea
	* [[Infinitude of Poulet numbers]]		\|-
			\| [[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 \|\| 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 \|\| 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>.
			\|-
			\| [[Infinitude of Poulet numbers]] \|\| \|\| Use the Mersenne number iteratively, after having found at least one Poulet number.
			\|}

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Facts */

2012-01-02T04:16:00Z

Facts

← Older revision		Revision as of 04:16, 2 January 2012
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	* [[Mersenne number for prime or Poulet implies prime or Poulet]]		* [[Mersenne number for prime or Poulet implies prime or Poulet]]
			* [[Composite Fermat number implies Poulet number]]
	* [[Infinitude of Poulet numbers]]		* [[Infinitude of Poulet numbers]]