7 - Revision history

Vipul: /* Properties and families */

2012-07-03T17:35:54Z

Properties and families

← Older revision		Revision as of 17:35, 3 July 2012
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	\|-		\|-
	\| [[satisfies property::twin primes]] (greater member of pair) \|\| second greater member twin prime \|\| {{#lst:twin primes\|list-greater}} \|\| 7 - 2 = [[5]] is prime		\| [[satisfies property::twin primes]] (greater member of pair) \|\| second greater member twin prime \|\| {{#lst:twin primes\|list-greater}} \|\| 7 - 2 = [[5]] is prime
			\|-
			\| [[satisfies property::Euclid number]] (number of the form [[primorial]] + 1) \|\| 1 more than the second primorial \|\| {{#lst:Euclid number\|list}} \|\| <math>7 = (2 \cdot 3) + 1</math>
			\|-
			\| [[satisfies property::factorial prime]] (prime that is a factorial <math>\pm 1</math>) \|\| <math>3! + 1</math> \|\| {{#lst:factorial prime\|list}} \|\| <math>7 = 3! + 1</math>
	\|}		\|}

Vipul: /* Properties and families */

2012-06-23T01:20:24Z

Properties and families

← Older revision		Revision as of 01:20, 23 June 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]] \|\| fourth prime number \|\| {{#lst:prime number\|list}} \|\| divide and check		\| [[satisfies property::prime number]] \|\| fourth prime number \|\| {{#lst:prime number\|list}} \|\| {{divide and check up to sqrt}} In this case, since <math>\sqrt{7}</math> is between 2 and 3, verifying primality requires verifying that 7 is not divisible by any prime up to 2, i.e., it is not divisible by [[2]].
	\|-		\|-
	\| [[satisfies property::Mersenne number]] <math>M_n = 2^n - 1</math> \|\| <math>n = 3</math>, i.e., <math>M_3 = 2^3 - 1</math> \|\| {{#lst:Mersenne number\|list}} \|\| plug and check		\| [[satisfies property::Mersenne number]] <math>M_n = 2^n - 1</math> \|\| <math>n = 3</math>, i.e., <math>M_3 = 2^3 - 1</math> \|\| {{#lst:Mersenne number\|list}} \|\| plug and check

Vipul: /* Properties and families */

2012-06-23T00:52:56Z

Properties and families

← Older revision		Revision as of 00:52, 23 June 2012
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	\| [[satisfies property::regular prime]] \|\| third regular prime \|\| {{#lst:regular prime\|list}} \|\|		\| [[satisfies property::regular prime]] \|\| third regular prime \|\| {{#lst:regular prime\|list}} \|\|
	\|-		\|-
	\| [[satisfies property::twin primes]] (greater member of pair) \|\| second greater member twin prime \|\| {{#lst:twin ~~prime~~\|list-greater}} \|\| 7 - 2 = [[5]] is prime		\| [[satisfies property::twin primes]] (greater member of pair) \|\| second greater member twin prime \|\| {{#lst:twin primes\|list-greater}} \|\| 7 - 2 = [[5]] is prime
	\|}		\|}

Vipul: /* Properties and families */

2012-06-23T00:52:43Z

Properties and families

← Older revision		Revision as of 00:52, 23 June 2012
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	\| [[satisfies property::regular prime]] \|\| third regular prime \|\| {{#lst:regular prime\|list}} \|\|		\| [[satisfies property::regular prime]] \|\| third regular prime \|\| {{#lst:regular prime\|list}} \|\|
	\|-		\|-
	\| [[satisfies property::twin ~~prime~~]] (~~bigger~~ member of pair) \|\| ~~third twin prime,~~ second ~~bigger~~ member twin prime \|\| {{#lst:twin prime\|list}} \|\| 7 - 2 = [[5]] is prime		\| [[satisfies property::twin primes]] (greater member of pair) \|\| second greater member twin prime \|\| {{#lst:twin prime\|list-greater}} \|\| 7 - 2 = [[5]] is prime
	\|}		\|}

Vipul: /* Orders of elements */

2012-01-03T21:27:17Z

Orders of elements

← Older revision		Revision as of 21:27, 3 January 2012
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	===Orders of elements===		===Orders of elements===

			The order of <math>a</math> mod 7 is the smallest positive integer <math>m</math> such that <math>a^m \equiv 1 \pmod 7</math>.

	The order of any element of the multiplicative group is the order of its discrete log in the corresponding additive group, i.e., we have:		The order of any element of the multiplicative group is the order of its discrete log in the corresponding additive group, i.e., we have:

Vipul: /* Structure of integers mod 7 */

2012-01-03T21:26:29Z

Structure of integers mod 7

@@ Line 34: / Line 34: @@
 {| class="sortable" border="1"
-! Congruence class mod 7 (written as smallest positive integer) !! Congruence class mod 7 (written as smallest magnitude integer) !! Discrete logarithm to base 3, written as integer mod 6 !! Is it a [[primitive root]] mod 7 (if and only if the discrete log is relatively prime to 6)? !! Is it s quadratic residue or nonresidue mod 7 (residue if discrete log is even, nonresidue if odd)
+! Congruence class mod 7 (written as smallest positive integer) !! Congruence class mod 7 (written as smallest magnitude integer) !! Discrete logarithm to base 3, written as integer mod 6 !! Is it a [[primitive root]] mod 7 (if and only if the discrete log is relatively prime to 6)? !! Is it a quadratic residue or nonresidue mod 7 (residue if discrete log is even, nonresidue if odd)
 |-
 | 1 || 1 || 0 || No || [[quadratic residue]]
@@ Line 69: / Line 69: @@
 |-
 ! -- !! --!! --!! Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] !! Total: 3 residues, 3 nonresidues. Equal number of both
 |}

Vipul: /* Discrete logarithm */

2012-01-03T20:10:24Z

Discrete logarithm

← Older revision		Revision as of 20:10, 3 January 2012
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	\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]		\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]
	\|-		\|-
	\| -- \|\| -- \|\| -- \|\| Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] \|\| Total: 3 residues, 3 nonresidues. Equal number of both		! -- !! -- !! -- !! Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] !! Total: 3 residues, 3 nonresidues. Equal number of both
	\|}		\|}

Vipul: /* Discrete logarithm */

2012-01-03T20:09:52Z

Discrete logarithm

← Older revision		Revision as of 20:09, 3 January 2012
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	\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]		\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]
	\|-		\|-
	! !! !! !! Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] !! Total: 3 residues, 3 nonresidues. Equal number of both		! -- !! --!! --!! Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] !! Total: 3 residues, 3 nonresidues. Equal number of both
	\|}		\|}

Vipul: /* Discrete logarithm */

2012-01-03T20:09:22Z

Discrete logarithm

← Older revision		Revision as of 20:09, 3 January 2012
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Congruence class mod 7 (written as smallest positive integer) !! Congruence class mod 7 (written as smallest magnitude integer) !! Discrete logarithm to base 5, written as integer mod 6 !! Is it a [[primitive root]] mod 7 (if and only if the discrete log is relatively prime to 6)? !! Is it s quadratic residue or nonresidue mod 7 (residue if discrete log is even, nonresidue if odd)		! Congruence class mod 7 (written as smallest positive integer) !! Congruence class mod 7 (written as smallest magnitude integer) !! Discrete logarithm to base 5, written as integer mod 6 !! Is it a [[primitive root]] mod 7 (if and only if the discrete log is relatively prime to 6)? !! Is it a quadratic residue or nonresidue mod 7 (residue if discrete log is even, nonresidue if odd)
	\|-		\|-
	\| 1 \|\| 1 \|\| 0 \|\| No \|\| [[quadratic residue]]		\| 1 \|\| 1 \|\| 0 \|\| No \|\| [[quadratic residue]]
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	\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]		\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]
	\|-		\|-
	~~\| \|\| \|\| \|\|~~ Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] \|\| Total: 3 residues, 3 nonresidues. Equal number of both		! !! !! !! Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] !! Total: 3 residues, 3 nonresidues. Equal number of both
	\|}		\|}

Vipul: /* Discrete logarithm */

2012-01-03T20:08:37Z

Discrete logarithm

← Older revision		Revision as of 20:08, 3 January 2012
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	\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]		\| 6 \|\| -1 \|\| 3 \|\| No \|\| [[quadratic nonresidue]]
	\|-		\|-
	\| ~~N/A~~ \|\| ~~N/A~~ \|\| ~~N/A~~ \|\| Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] \|\| Total: 3 residues, 3 nonresidues. Equal number of both		\| \|\| \|\| \|\| Total: 2 Yes, 4 No<br><math>2 = \varphi(6)</math>, <math>\varphi</math> is [[Euler totient function]] \|\| Total: 3 residues, 3 nonresidues. Equal number of both
	\|}		\|}