17 - Revision history

Vipul: /* Properties and families */

2013-02-22T06:28:33Z

Properties and families

← Older revision		Revision as of 06:28, 22 February 2013
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	! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment		! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
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	\| [[satisfies property::prime number]] \|\| it is the 7th prime number \|\| {{#lst:prime number\|list}} \|\| {{divide and check up to sqrt}} Since <math>\sqrt{13}</math> is between [[4]] and [[5]], we only need to check divisibility by primes less than or equal to 4, i.e., we need to verify that 13 is not divisible by the primes [[2]] and [[3]].		\| [[satisfies property::prime number]] \|\| it is the 7th prime number \|\| {{#lst:prime number\|list}} \|\| {{divide and check up to sqrt}} Since <math>\sqrt{17}</math> is between [[4]] and [[5]], we only need to check divisibility by primes less than or equal to 4, i.e., we need to verify that 17 is not divisible by the primes [[2]] and [[3]].
	\|-		\|-
	\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check		\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check

Vipul: /* Properties and families */

2012-07-03T18:26:31Z

Properties and families

← Older revision		Revision as of 18:26, 3 July 2012
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	\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|		\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|
	\|-		\|-
	\| [[satisfies property::near-square prime]] of the form <math>n^2 + 1</math> \|\| <math>n = 4</math> (third prime of this form) \|\| {{#lst:near-square prime\|plus-one-list}}		\| [[satisfies property::near-square prime]] of the form <math>n^2 + 1</math> \|\| <math>n = 4</math> (third prime of this form) \|\| {{#lst:near-square prime\|plus-one-list}} \|\|
	\|-		\|-
	\| [[satisfies property::lucky number of Euler]] \|\| fifth of six such numbers \|\| {{#lst:lucky number of Euler\|list}} \|\| A prime <math>p</math> is a lucky number of Euler iff the ring of integers in <math>\mathbb{Q}(\sqrt{1 - 4p})</math> is a unique factorization domain.		\| [[satisfies property::lucky number of Euler]] \|\| fifth of six such numbers \|\| {{#lst:lucky number of Euler\|list}} \|\| A prime <math>p</math> is a lucky number of Euler iff the ring of integers in <math>\mathbb{Q}(\sqrt{1 - 4p})</math> is a unique factorization domain.

Vipul: /* Properties and families */

2012-07-03T18:26:21Z

Properties and families

← Older revision		Revision as of 18:26, 3 July 2012
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	\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|		\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|
			\|-
			\| [[satisfies property::near-square prime]] of the form <math>n^2 + 1</math> \|\| <math>n = 4</math> (third prime of this form) \|\| {{#lst:near-square prime\|plus-one-list}}
	\|-		\|-
	\| [[satisfies property::lucky number of Euler]] \|\| fifth of six such numbers \|\| {{#lst:lucky number of Euler\|list}} \|\| A prime <math>p</math> is a lucky number of Euler iff the ring of integers in <math>\mathbb{Q}(\sqrt{1 - 4p})</math> is a unique factorization domain.		\| [[satisfies property::lucky number of Euler]] \|\| fifth of six such numbers \|\| {{#lst:lucky number of Euler\|list}} \|\| A prime <math>p</math> is a lucky number of Euler iff the ring of integers in <math>\mathbb{Q}(\sqrt{1 - 4p})</math> is a unique factorization domain.

Vipul: /* Properties and families */

2012-07-03T18:10:05Z

Properties and families

← Older revision		Revision as of 18:10, 3 July 2012
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	! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment		! Property or family !! Parameter values !! First few numbers !! Proof of satisfaction/membership/containment
	\|-		\|-
	\| [[satisfies property::prime number]] \|\| it is the 7th prime number \|\| {{#lst:prime number\|list}} \|\| divide and check		\| [[satisfies property::prime number]] \|\| it is the 7th prime number \|\| {{#lst:prime number\|list}} \|\| {{divide and check up to sqrt}} Since <math>\sqrt{13}</math> is between [[4]] and [[5]], we only need to check divisibility by primes less than or equal to 4, i.e., we need to verify that 13 is not divisible by the primes [[2]] and [[3]].
	\|-		\|-
	\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check		\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check

Vipul: /* Properties and families */

2012-07-03T18:07:11Z

Properties and families

← Older revision		Revision as of 18:07, 3 July 2012
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	\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check		\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check
	\|-		\|-
	\| [[satisfies property::Proth prime]] ~~\|\| Prime~~ of the form <math>k \cdot 2^n + 1</math> with <math>2^n > k</math> ~~\|\| {{#lst:Proth prime\|list}}~~ \|\| <math>n = 4, k = 1</math>		\| [[satisfies property::Proth prime]]: prime of the form <math>k \cdot 2^n + 1</math> with <math>2^n > k</math> \|\| <math>n = 4, k = 1</math> \|\| {{#lst:Proth prime\|list}} \|\|
	\|-		\|-
	\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|		\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|

Vipul: /* Properties and families */

2012-07-03T18:06:49Z

Properties and families

← Older revision		Revision as of 18:06, 3 July 2012
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	\|-		\|-
	\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check		\| [[satisfies property::Fermat number]], [[satisfies property::Fermat prime]] \|\| <math>F_2</math>, where <math>F_n = 2^{2^n} + 1</math>, starts <math>n = 0</math> \|\| {{#lst:Fermat number\|list}} \|\| plug and check
			\|-
			\| [[satisfies property::Proth prime]] \|\| Prime of the form <math>k \cdot 2^n + 1</math> with <math>2^n > k</math> \|\| {{#lst:Proth prime\|list}} \|\| <math>n = 4, k = 1</math>
	\|-		\|-
	\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|		\| [[satisfies property::regular prime]] \|\| sixth regular prime (2 is neither regular nor irregular) \|\| {{#lst:regular prime\|list}} \|\|

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

2012-07-03T17:44:56Z

Irreducible polynomials by Cohn's irreducibility criterion

← Older revision		Revision as of 17:44, 3 July 2012
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	===Irreducible polynomials by Cohn's irreducibility criterion===		===Irreducible polynomials by Cohn's irreducibility criterion===

	By [[Cohn's irreducibility criterion]], we know that if we write 17 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here the ~~first few~~ irreducible polynomials:		By [[Cohn's irreducibility criterion]], we know that if we write 17 in any base greater than or equal to 2, the corresponding polynomial is irreducible. We list here all the irreducible polynomials:

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

Vipul: /* Prime-generating polynomials */

2012-07-03T17:28:08Z

Prime-generating polynomials

@@ Line 172: / Line 172: @@
 is a [[Fermat prime]], hence the regular 17-gon is constructible by straightedge and compass. This is because the cyclotomic extension of adjoining 17th roots of unity can be obtained by taking successive quadratic extensions.
-==Prime-generating polynomials==
+==Polynomials==
 Below are some polynomials that give prime numbers for small input values, which give the value 17 for suitable input choice.
@@ Line 184: / Line 186: @@
 |-
 | <math>n^2 - n + 17</math> || 2 || all numbers 1-16, because 17 is one of the [[lucky numbers of Euler]]. || 1
 |}

Vipul at 17:23, 3 July 2012

2012-07-03T17:23:55Z

← Older revision		Revision as of 17:23, 3 July 2012
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			{{particular natural number}}

	==Summary==		==Summary==

Vipul: /* Condition for 17 to be a quadratic residue modulo a prime */

2012-05-21T02:20:18Z

Condition for 17 to be a quadratic residue modulo a prime

← Older revision		Revision as of 02:20, 21 May 2012
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	Whether or not 17 is a quadratic residue modulo a prime depends ''only'' on the congruence class of the prime modulo 68 (obtainedas 4 times 17). This follows from [[quadratic reciprocity]] for odd primes.		Whether or not 17 is a quadratic residue modulo a prime depends ''only'' on the congruence class of the prime modulo 68 (obtainedas 4 times 17). This follows from [[quadratic reciprocity]] for odd primes.

	By [[Dirichlet's theorem on primes in arithmetic progressions]], we thus find that there are infinitely many primes <math>p</math> for which 7 is a quadratic residue and infinitely many primes <math>p</math> for which 7 is a quadratic nonresidue.		By [[Dirichlet's theorem on primes in arithmetic progressions]], we thus find that there are infinitely many primes <math>p</math> for which 17 is a quadratic residue and infinitely many primes <math>p</math> for which 17 is a quadratic nonresidue.

	===Square roots of -1===		===Square roots of -1===