Twin primes - Revision history

Vipul: /* Occurrence */

2012-06-23T00:51:56Z

Occurrence

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	===Lesser of twin primes===		===Lesser of twin primes===

	The ''lesser of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p + 2</math> is prime. This list goes: <section begin="list-lesser"/>[[3]], [[5]], [[11]], [[17]], [[29]], [[41]], [[59]], [[71]], [[101]], <toggledisplay>107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607</toggledisplay>{{oeis\|~~A001359}}~~<section end="list-lesser"/>		The ''lesser of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p + 2</math> is prime. This list goes: <section begin="list-lesser"/>[[3]], [[5]], [[11]], [[17]], [[29]], [[41]], [[59]], [[71]], [[101]], <toggledisplay>107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607</toggledisplay>[[oeis:A001359\|View list on OEIS]]<section end="list-lesser"/>

	===Greater of twin primes===		===Greater of twin primes===

	The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: <section begin="list-greater"/>[[5]], [[7]], [[13]], [[19]], [[31]], [[43]], [[61]], [[73]], [[103]], <toggledisplay>109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609</toggledisplay>~~{{oeis~~\|~~A006512}}~~<section end="list-greater"/>		The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: <section begin="list-greater"/>[[5]], [[7]], [[13]], [[19]], [[31]], [[43]], [[61]], [[73]], [[103]], <toggledisplay>109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609</toggledisplay>[[Oeis:A006512\|View list on OEIS]]<section end="list-greater"/>

	===Proportion of primes===		===Proportion of primes===

Vipul: /* Occurrence */

2012-06-23T00:50:43Z

Occurrence

← Older revision		Revision as of 00:50, 23 June 2012
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	===Lesser of twin primes===		===Lesser of twin primes===

	~~{{oeis\|A001359}}~~		The ''lesser of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p + 2</math> is prime. This list goes: <section begin="list-lesser"/>[[3]], [[5]], [[11]], [[17]], [[29]], [[41]], [[59]], [[71]], [[101]], <toggledisplay>107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607</toggledisplay>{{oeis\|A001359}}<section end="list-lesser"/>

	The ''lesser of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p + 2</math> is prime. This list goes: 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607.

	===Greater of twin primes===		===Greater of twin primes===

	~~{{oeis\|A006512}}~~		The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: <section begin="list-greater"/>[[5]], [[7]], [[13]], [[19]], [[31]], [[43]], [[61]], [[73]], [[103]], <toggledisplay>109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609</toggledisplay>{{oeis\|A006512}}<section end="list-greater"/>

	The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609.

	===Proportion of primes===		===Proportion of primes===

Vipul: /* Proportion of primes */

2010-05-03T20:23:39Z

Proportion of primes

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	{\| class="wikitable" border="1"		{\| class="wikitable" border="1"
	! <math>n</math> !! Number of primes <math>\le n</math> !! Number of twin prime pairs <math>\le n</math> !! Proportion of primes that are members of twin prime pairs !!		! <math>n</math> !! Number of primes <math>\le n</math> !! Number of twin prime pairs <math>\le n</math> !! Proportion of primes that are members of twin prime pairs
	\|-		\|-
	\| 10 \|\| 4 \|\| 2 \|\| <math>3/4 = 0.75</math> \|\|		\| 10 \|\| 4 \|\| 2 \|\| <math>3/4 = 0.75</math>
	\|-		\|-
	\| 100 \|\| 25 \|\| 8 \|\| <math>15/25 = 0.60</math> \|\|		\| 100 \|\| 25 \|\| 8 \|\| <math>15/25 = 0.60</math>
	\|-		\|-
	\| 1000 \|\| 168 \|\| 35 \|\| <math>69/168 \approx 0.410714</math> \|\|		\| 1000 \|\| 168 \|\| 35 \|\| <math>69/168 \approx 0.410714</math>
	\|-		\|-
	\| 10000 \|\| 1229 \|\| 205 \|\| <math>409/1229 \~~approc~~ 0.332791</math> \|\|		\| 10000 \|\| 1229 \|\| 205 \|\| <math>409/1229 \approx 0.332791</math>
	\|}		\|}

Vipul: /* Particular cases */

2010-05-03T20:23:01Z

Particular cases

← Older revision		Revision as of 20:23, 3 May 2010
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	* In particular, with the exception of the prime <math>5</math>, any member of a pair of twin primes cannot be a member of ''another'' pair of twin primes, i.e., the lesser member in one pair of twin primes cannot be the greater member in another pair.		* In particular, with the exception of the prime <math>5</math>, any member of a pair of twin primes cannot be a member of ''another'' pair of twin primes, i.e., the lesser member in one pair of twin primes cannot be the greater member in another pair.

	==~~Particular cases~~==		==Occurrence==

	===Lesser of twin primes===		===Lesser of twin primes===
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	The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609.		The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609.

			===Proportion of primes===

			{\| class="wikitable" border="1"
			! <math>n</math> !! Number of primes <math>\le n</math> !! Number of twin prime pairs <math>\le n</math> !! Proportion of primes that are members of twin prime pairs !!
			\|-
			\| 10 \|\| 4 \|\| 2 \|\| <math>3/4 = 0.75</math> \|\|
			\|-
			\| 100 \|\| 25 \|\| 8 \|\| <math>15/25 = 0.60</math> \|\|
			\|-
			\| 1000 \|\| 168 \|\| 35 \|\| <math>69/168 \approx 0.410714</math> \|\|
			\|-
			\| 10000 \|\| 1229 \|\| 205 \|\| <math>409/1229 \approc 0.332791</math> \|\|
			\|}

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

Vipul at 20:00, 3 May 2010

2010-05-03T20:00:06Z

← Older revision		Revision as of 20:00, 3 May 2010
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	The [[twin prime conjecture]] states that there are infinitely many twin primes.		The [[twin prime conjecture]] states that there are infinitely many twin primes.

			==Basic facts==

			* If <math>p</math> and <math>p + 2</math> are both prime, they both ''must'' be odd numbers.
			* For <math>p > 3</math>, if <math>(p,p+2)</math> are twin primes, then <math>p \equiv 2 \pmod 3</math> (hence <math>p \equiv 5 \pmod 6</math>) and <math>p + 2 \equiv 1 \pmod 3</math> (hence <math>p + 2 \equiv 1 \pmod 6</math>).
			* In particular, with the exception of the prime <math>5</math>, any member of a pair of twin primes cannot be a member of ''another'' pair of twin primes, i.e., the lesser member in one pair of twin primes cannot be the greater member in another pair.

	==Particular cases==		==Particular cases==

Vipul: /* Lesser of twin primes= */

2010-05-03T19:47:19Z

Lesser of twin primes=

← Older revision		Revision as of 19:47, 3 May 2010
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	==Particular cases==		==Particular cases==

	==Lesser of twin primes===		===Lesser of twin primes===

	{{oeis\|A001359}}		{{oeis\|A001359}}
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	The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609.		The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609.


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

Vipul at 19:46, 3 May 2010

2010-05-03T19:46:13Z

← Older revision		Revision as of 19:46, 3 May 2010
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	The [[twin prime conjecture]] states that there are infinitely many twin primes.		The [[twin prime conjecture]] states that there are infinitely many twin primes.

			==Particular cases==

			==Lesser of twin primes===

			{{oeis\|A001359}}

			The ''lesser of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p + 2</math> is prime. This list goes: 3, 5, 11, 17, 29, 41, 59, 71, 101, 107, 137, 149, 179, 191, 197, 227, 239, 269, 281, 311, 347, 419, 431, 461, 521, 569, 599, 617, 641, 659, 809, 821, 827, 857, 881, 1019, 1031, 1049, 1061, 1091, 1151, 1229, 1277, 1289, 1301, 1319, 1427, 1451, 1481, 1487, 1607.

			===Greater of twin primes===

			{{oeis\|A006512}}

			The ''greater of twin primes'' is the sequence (in increasing order) of all the primes <math>p</math> for which <math>p - 2</math> is prime. This list goes: 5, 7, 13, 19, 31, 43, 61, 73, 103, 109, 139, 151, 181, 193, 199, 229, 241, 271, 283, 313, 349, 421, 433, 463, 523, 571, 601, 619, 643, 661, 811, 823, 829, 859, 883, 1021, 1033, 1051, 1063, 1093, 1153, 1231, 1279, 1291, 1303, 1321, 1429, 1453, 1483, 1489, 1609.


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

Vipul: /* Relation with other properties */

2010-05-02T02:16:09Z

Relation with other properties

← Older revision		Revision as of 02:16, 2 May 2010
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	\|-		\|-
	\| [[Prime constellation]] \|\| a sequence of consecutive primes <math>p_1 < p_2 < \dots < p_k</math> for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes \|\| we are usually interested in prime constellation having a particular constellation pattern.		\| [[Prime constellation]] \|\| a sequence of consecutive primes <math>p_1 < p_2 < \dots < p_k</math> for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes \|\| we are usually interested in prime constellation having a particular constellation pattern.
			\|-
			\| [[Bitwin chain]] \|\| \|\| combines the idea of twin primes, [[Cunningham chain of the first kind]], and [[Cunningham chain of the second kind]]
	\|}		\|}

Vipul at 02:11, 2 May 2010

2010-05-02T02:11:27Z

← Older revision		Revision as of 02:11, 2 May 2010
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	==Definition==		==Definition==

	The term '''twin primes''' is used for a pair of odd prime numbers that differ by two. In other words, primes <math>p, p + 2</math> are termed twin primes.		The term '''twin primes''' is used for a pair of odd prime numbers that differ by two. In other words, primes <math>p, p + 2</math> are termed twin primes. Either member of a pair of twin primes may be referred to as a '''twin prime'''.

	The [[twin ~~primes~~ conjecture]] states that there are infinitely many twin primes.		The [[twin prime conjecture]] states that there are infinitely many twin primes.

	==Relation with other properties==		==Relation with other properties==
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	===Related properties for pairs of primes===		===Related properties for pairs of primes===

	* [[Cousin primes]] ~~are~~ two primes that differ by <math>4</math>. Note that for <math>p > 3</math>, if both <math>p</math> and <math>p + 4</math> are prime, <math>p+2</math> is not prime.		{\| class="wikitable" border="1"
	* [[Sexy primes]] ~~are~~ two primes that differ by <math>6</math> (~~and~~ with no prime in between).		! Property !! Meaning !! Comment
			\|-
			\| [[Cousin primes]] \|\| two primes that differ by <math>4</math> \|\| Note that for <math>p > 3</math>, if both <math>p</math> and <math>p + 4</math> are prime, <math>p+2</math> is not prime. Hence, the prime gap in this case is <math>4</math>.
			\|-
			\| [[Sexy primes]] \|\| two primes that differ by <math>6</math> (with no prime in between) \|\| Since this is a pair of successive primes, the prime gap is <math>6</math>.
			\|-
			\| [[Sophie Germain prime]] \|\| a prime <math>p</math> such that <math>2p + 1</math> is also prime \|\| the corresponding prime <math>2p + 1</math> is a safe prime
			\|-
			\| [[safe prime]] \|\| a prime <math>p</math> such that <math>(p - 1)/2</math> is also prime \|\| the corresponding prime <math>(p - 1)/2</math> is a Sophie Germain prime
			\|}

			===Related properties for primes===

			* [[Chen prime]] is a [[prime number]] <math>p</math> such that <math>p + 2</math> is either a prime number or a [[semiprime]]. The name arises because of [[Chen's theorem on primes and semiprimes with fixed separation]], which essentially asserts that for any fixed separation, there are infinitely many pairs of a prime and a semiprime having that separation.
	===Related properties for more than two primes===		===Related properties for more than two primes===

	* [[Prime quadruplet]] is a collection of four primes <math>p,p+2,p+6,p+8</math>.		{\| class="wikitable" border="1"
	* [[Prime constellation]] is a sequence of consecutive primes <math>p_1 < p_2 < \dots < p_k</math> for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes.		! Property !! Meaning !! Comment
			\|-
			\| [[Prime quadruplet]] \|\| a collection of four primes <math>p,p+2,p+6,p+8</math> \|\| there can be no further primes in between
			\|-
			\| [[Prime constellation]] \|\| a sequence of consecutive primes <math>p_1 < p_2 < \dots < p_k</math> for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes \|\| we are usually interested in prime constellation having a particular constellation pattern.
			\|}

			==Related facts/conjectures==

			{\| class="wikitable" border="1"
			! Broad concern !! Name of fact/conjecture !! Statement !! Status
			\|-
			\| Infinitude \|\| [[twin primes conjecture]] \|\| there are infinitely many twin primes \|\| open
			\|-
			\| Largeness, i.e., sum of reciprocals \|\| [[Brun's theorem]] \|\| the sum of reciprocals of all twin prime pairs (i.e., we add the reciprocal of each member of each twin prime pair) is finite \|\| proved. This sum is [[Brun's constant]].
			\|-
			\| Density \|\| [[first Hardy-Littlewood conjecture]] \|\| In the particular case of twin primes, the claim is that the number of twin prime pairs <math>\le n</math> is <math>\! \sim 2C_2n/(\ln n)^2</math>, where <math>C_2</math> is a specified constant.
			\|}

Vipul at 18:35, 20 April 2009

2009-04-20T18:35:30Z

← Older revision		Revision as of 18:35, 20 April 2009
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	The [[twin primes conjecture]] states that there are infinitely many twin primes.		The [[twin primes conjecture]] states that there are infinitely many twin primes.

			==Relation with other properties==

			===Related properties for pairs of primes===

			* [[Cousin primes]] are two primes that differ by <math>4</math>. Note that for <math>p > 3</math>, if both <math>p</math> and <math>p + 4</math> are prime, <math>p+2</math> is not prime.
			* [[Sexy primes]] are two primes that differ by <math>6</math> (and with no prime in between).

			===Related properties for more than two primes===

			* [[Prime quadruplet]] is a collection of four primes <math>p,p+2,p+6,p+8</math>.
			* [[Prime constellation]] is a sequence of consecutive primes <math>p_1 < p_2 < \dots < p_k</math> for which the difference between the first and last prime is the least possible based on considerations of modular arithmetic relative to smaller primes.