23 - Revision history

Vipul: /* Irreducible polynomials by Cohn's irreducibility criterion */

2012-07-03T17:46:00Z

Irreducible polynomials by Cohn's irreducibility criterion

← Older revision		Revision as of 17:46, 3 July 2012
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Base <math>b</math> !! 17 in base <math>b</math> !! Corresponding irreducible polynomial		! Base <math>b</math> !! 23 in base <math>b</math> !! Corresponding irreducible polynomial
	\|-		\|-
	\| 2 \|\| 10111 \|\| <math>x^4 + x^2 + x + 1</math>		\| 2 \|\| 10111 \|\| <math>x^4 + x^2 + x + 1</math>

Vipul: /* Prime-generating polynomials */

2012-07-03T17:45:47Z

Prime-generating polynomials

@@ Line 21: / Line 21: @@
 |}
-==Prime-generating polynomials==
+==Polynomials==
 Below are some polynomials that give prime numbers for small input values, which give the value 23 for suitable input choice.
@@ Line 31: / Line 33: @@
 |-
 | <math>n^2 - n + 17</math> || 2 || all numbers 1-16, because 17 is one of the [[lucky numbers of Euler]]. || 3
 |}

Vipul: /* Properties and families */

2012-07-03T17:43:18Z

Properties and families

← Older revision		Revision as of 17:43, 3 July 2012
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	\| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) \|\| fifth Sophie Germain prime \|\| {{#lst:Sophie Germain prime\|list}} \|\| <math>2(23) + 1</math> equals [[47]], which is prime		\| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) \|\| fifth Sophie Germain prime \|\| {{#lst:Sophie Germain prime\|list}} \|\| <math>2(23) + 1</math> equals [[47]], which is prime
	\|-		\|-
	\| [[satisfies property::factorial prime]] (prime that is of the form [[factorial]] <math>\pm 1</math>) \|\| fifth factorial prime \|\| {{#lst:factorial prime\|list}} \|\| <math>23 = 4! ~~- 1~~</math>		\| [[satisfies property::factorial prime]] (prime that is of the form [[factorial]] <math>\pm 1</math>) \|\| fifth factorial prime \|\| {{#lst:factorial prime\|list}} \|\| <math>23 = 24 - 1</math> and [[24]] equals <math>4!</math>
	\|}		\|}

Vipul: /* Properties and families */

2012-07-03T17:42:59Z

Properties and families

← Older revision		Revision as of 17:42, 3 July 2012
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	\| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) \|\| fifth Sophie Germain prime \|\| {{#lst:Sophie Germain prime\|list}} \|\| <math>2(23) + 1</math> equals [[47]], which is prime		\| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) \|\| fifth Sophie Germain prime \|\| {{#lst:Sophie Germain prime\|list}} \|\| <math>2(23) + 1</math> equals [[47]], which is prime
	\|-		\|-
	\| [[satisfies property::factorial prime]] (prime that is of the form [[factorial]] <math>\pm 1</math> \|\| fifth factorial prime \|\| {{#lst:factorial prime\|list}} \|\| <math>23 = 4! - 1</math>		\| [[satisfies property::factorial prime]] (prime that is of the form [[factorial]] <math>\pm 1</math>) \|\| fifth factorial prime \|\| {{#lst:factorial prime\|list}} \|\| <math>23 = 4! - 1</math>
	\|}		\|}

Vipul: /* Properties and families */

2012-07-03T17:42:46Z

Properties and families

← Older revision		Revision as of 17:42, 3 July 2012
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	\| [[satisfies property::prime number]] \|\| ninth prime number \|\| {{#lst:prime number\|list}} \|\| {{divide and check up to sqrt}} In this case, since <math>\sqrt{23}</math> is between 4 and 5, verifying primality requires verifying that 23 is not divisible by any prime up to 4, i.e., it is not divisible by [[2]] or [[3]].		\| [[satisfies property::prime number]] \|\| ninth prime number \|\| {{#lst:prime number\|list}} \|\| {{divide and check up to sqrt}} In this case, since <math>\sqrt{23}</math> is between 4 and 5, verifying primality requires verifying that 23 is not divisible by any prime up to 4, i.e., it is not divisible by [[2]] or [[3]].
	\|-		\|-
	\| [[satisfies property::safe prime]] (prime <math>p</math> such that <math>(p - 1)/2</math> is prime) \|\| \|\| {{#lst:safe prime\|list}} \|\| <math>(23 - 1)/2</matH> equals [[11]], which is prime		\| [[satisfies property::safe prime]] (prime <math>p</math> such that <math>(p - 1)/2</math> is prime) \|\| fourth safe prime \|\| {{#lst:safe prime\|list}} \|\| <math>(23 - 1)/2</matH> equals [[11]], which is prime
	\|-		\|-
	\| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) \|\| \|\| {{#lst:Sophie Germain prime\|list}} \|\| <math>2(23) + 1</math> equals [[47]], which is prime		\| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) \|\| fifth Sophie Germain prime \|\| {{#lst:Sophie Germain prime\|list}} \|\| <math>2(23) + 1</math> equals [[47]], which is prime
	\|-		\|-
	\| [[satisfies property::factorial prime]] (prime that is of the form [[factorial]] <math>\pm 1</math> \|\| \|\| {{#lst:factorial prime\|list}} \|\| <math>23 = 4! - 1</math>		\| [[satisfies property::factorial prime]] (prime that is of the form [[factorial]] <math>\pm 1</math> \|\| fifth factorial prime \|\| {{#lst:factorial prime\|list}} \|\| <math>23 = 4! - 1</math>
	\|}		\|}

	==Prime-generating polynomials==		==Prime-generating polynomials==

Vipul at 17:42, 3 July 2012

2012-07-03T17:42:17Z

← Older revision		Revision as of 17:42, 3 July 2012
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	The number 23 is a [[prime number]].		The number 23 is a [[prime number]].

			===Properties and families===

			{\| class="sortable" border="1"
			! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
			\|-
			\| [[satisfies property::prime number]] \|\| ninth prime number \|\| {{#lst:prime number\|list}} \|\| {{divide and check up to sqrt}} In this case, since <math>\sqrt{23}</math> is between 4 and 5, verifying primality requires verifying that 23 is not divisible by any prime up to 4, i.e., it is not divisible by [[2]] or [[3]].
			\|-
			\| [[satisfies property::safe prime]] (prime <math>p</math> such that <math>(p - 1)/2</math> is prime) \|\| \|\| {{#lst:safe prime\|list}} \|\| <math>(23 - 1)/2</matH> equals [[11]], which is prime
			\|-
			\| [[satisfies property::Sophie Germain prime]] (prime <math>p</math> such that <math>2p + 1</math> is prime) \|\| \|\| {{#lst:Sophie Germain prime\|list}} \|\| <math>2(23) + 1</math> equals [[47]], which is prime
			\|-
			\| [[satisfies property::factorial prime]] (prime that is of the form [[factorial]] <math>\pm 1</math> \|\| \|\| {{#lst:factorial prime\|list}} \|\| <math>23 = 4! - 1</math>
			\|}
	==Prime-generating polynomials==		==Prime-generating polynomials==

Vipul at 21:49, 15 January 2012

2012-01-15T21:49:42Z

← Older revision		Revision as of 21:49, 15 January 2012
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	The number 23 is a [[prime number]].		The number 23 is a [[prime number]].

			==Prime-generating polynomials==

			Below are some polynomials that give prime numbers for small input values, which give the value 23 for suitable input choice.

			{\| class="sortable" border="1"
			! Polynomial !! Degree !! Some values for which it generates primes !! Input value <math>n</math> at which it generates 23
			\|-
			\| <math>n^2 - n + 11</math> \|\| 2 \|\| all numbers 1-10, because 11 is one of the [[lucky numbers of Euler]]. \|\| 4
			\|-
			\| <math>n^2 - n + 17</math> \|\| 2 \|\| all numbers 1-16, because 17 is one of the [[lucky numbers of Euler]]. \|\| 3
			\|}

Vipul: Created page with "{{particular natural number}} ==Summary== ===Factorization=== The number 23 is a prime number."

2012-01-02T22:55:23Z

Created page with "{{particular natural number}} ==Summary== ===Factorization=== The number 23 is a prime number."

New page

{{particular natural number}}

==Summary==

===Factorization===

The number 23 is a [[prime number]].