Twin prime conjecture - Revision history

Vipul: /* Relation with other conjectures and known facts */

2014-01-29T21:10:57Z

Relation with other conjectures and known facts

← Older revision		Revision as of 21:10, 29 January 2014
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	The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. For more results on the current state of the art in knowing of the limit inferior of prime gaps (both unconditional and conditional to various hypotheses) see the [[prime gap]] page.		The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. For more results on the current state of the art in knowing of the limit inferior of prime gaps (both unconditional and conditional to various hypotheses) see the [[prime gap]] page.

	~~===Average prime gap===~~

	~~The [[prime number theorem]] states that the average prime gap is the natural logarithm of the prime.~~

	~~===Supremum of prime gaps===~~

	* [[Bertrand's postulate]] (which is in fact a theorem) states that there always exists a prime between any number and its double.

	===Generalizations===		===Generalizations===

	* [[Schinzel's hypothesis H]] ~~is a much stronger and more general~~ conjecture ~~that provides a framework within which the twin primes~~ conjecture ~~fits~~.		* [[Polignac's conjecture]] states that every even number occurs infinitely often as a prime gap.
			* [[Dickson's conjecture]] generalizes the twin prime conjecture somewhat.
			* [[Schinzel's hypothesis H]] generalizes Dickson's conjecture.
			* [[Bateman-Horn conjecture]] generalizes Schinzel's Hypothesis H.

Vipul: /* Infimum of prime gaps */

2014-01-29T21:09:39Z

Infimum of prime gaps

← Older revision		Revision as of 21:09, 29 January 2014
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	==Relation with other conjectures and known facts==		==Relation with other conjectures and known facts==

	===~~Infimum~~ of prime gaps===		===Limit inferior of prime gaps===

	The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. ~~There are various~~ results ~~that prove bounds~~ on the limit inferior:		The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. For more results on the current state of the art in knowing of the limit inferior of prime gaps (both unconditional and conditional to various hypotheses) see the [[prime gap]] page.

	* [[Goldston-Pintz-Yildirim theorem on prime gaps conditional to ~~Elliott-Halberstam]]: The [[Elliott-Halberstam conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that~~ the ~~limit inferior of prime gaps is at most <math>16</math>.~~
	* [[~~Goldston-Pintz-Yildirim theorem on prime gaps relative to logarithm of~~ prime]]~~: The work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero~~.

	===Average prime gap===		===Average prime gap===

Vipul: /* Relation with other conjectures and known facts */

2012-07-03T01:03:23Z

Relation with other conjectures and known facts

← Older revision		Revision as of 01:03, 3 July 2012
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	* [[Bertrand's postulate]] (which is in fact a theorem) states that there always exists a prime between any number and its double.		* [[Bertrand's postulate]] (which is in fact a theorem) states that there always exists a prime between any number and its double.

			===Generalizations===

			* [[Schinzel's hypothesis H]] is a much stronger and more general conjecture that provides a framework within which the twin primes conjecture fits.

Vipul: moved Twin primes conjecture to Twin prime conjecture

2010-05-02T02:11:49Z

moved Twin primes conjecture to Twin prime conjecture

← Older revision	Revision as of 02:11, 2 May 2010
(No difference)

Vipul: /* Relation with other conjectures and known facts */

2010-02-09T03:17:24Z

Relation with other conjectures and known facts

← Older revision		Revision as of 03:17, 9 February 2010
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	The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. There are various results that prove bounds on the limit inferior:		The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. There are various results that prove bounds on the limit inferior:

	* [[Goldston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam]]: The [[Elliott-~~Halberstram~~ conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most <math>16</math>.		* [[Goldston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam]]: The [[Elliott-Halberstam conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most <math>16</math>.
	* [[Goldston-Pintz-Yildirim theorem on prime gaps relative to logarithm of prime]]: The work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.		* [[Goldston-Pintz-Yildirim theorem on prime gaps relative to logarithm of prime]]: The work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.

Vipul: /* Relation with other conjectures and known facts */

2010-02-09T03:17:06Z

Relation with other conjectures and known facts

← Older revision		Revision as of 03:17, 9 February 2010
Line 14:		Line 14:
	The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. There are various results that prove bounds on the limit inferior:		The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. There are various results that prove bounds on the limit inferior:

	* [[~~Golston~~-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam]]: The [[Elliott-Halberstram conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most <math>16</math>.		* [[Goldston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam]]: The [[Elliott-Halberstram conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most <math>16</math>.
	* [[~~Golston~~-Pintz-Yildirim theorem on prime gaps relative to logarithm of prime]]: The work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.		* [[Goldston-Pintz-Yildirim theorem on prime gaps relative to logarithm of prime]]: The work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.

	===Average prime gap===		===Average prime gap===

Vipul: /* Relation with other conjectures and known facts */

2010-02-09T03:15:08Z

Relation with other conjectures and known facts

← Older revision		Revision as of 03:15, 9 February 2010
Line 14:		Line 14:
	The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. There are various results that prove bounds on the limit inferior:		The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. There are various results that prove bounds on the limit inferior:

	* The [[Elliott-Halberstram conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most <math>16</math>.		* [[Golston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam]]: The [[Elliott-Halberstram conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most <math>16</math>.
	* ~~wThe~~ work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.		* [[Golston-Pintz-Yildirim theorem on prime gaps relative to logarithm of prime]]: The work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.

	===Average prime gap===		===Average prime gap===

Vipul at 22:30, 6 April 2009

2009-04-06T22:30:15Z

← Older revision		Revision as of 22:30, 6 April 2009
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	{{prime gap conjecture}}		{{prime gap conjecture}}
	{{infinitude conjecture}}		{{infinitude conjecture\|twin primes}}

	==Statement==		==Statement==

Vipul at 22:29, 6 April 2009

2009-04-06T22:29:22Z

← Older revision		Revision as of 22:29, 6 April 2009
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	{{prime gap conjecture}}		{{prime gap conjecture}}
			{{infinitude conjecture}}

	==Statement==		==Statement==

Vipul: Created page with '{{prime gap conjecture}} ==Statement== There are infinitely many fact about::twin primes. In other words, there are infinitely many odd primes $p$ such that
2009-04-06T22:18:15Z

Created page with '{{prime gap conjecture}} ==Statement== There are infinitely many fact about::twin primes. In other words, there are infinitely many odd primes <math>p</math> such that <mat...'

New page
{{prime gap conjecture}}

==Statement==

There are infinitely many [[fact about::twin primes]]. In other words, there are infinitely many odd primes <math>p</math> such that <math>p + 2</math> is also a prime.

In other words, the limit inferior of all [[prime gap]]s is <math>2</math>.

==Relation with other conjectures and known facts==

===Infimum of prime gaps===

The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is <math>2</math>. There are various results that prove bounds on the limit inferior:

* The [[Elliott-Halberstram conjecture]], by work of Dan Goldston, J´anos Pintz, and Cem Yıldırım, implies that the limit inferior of prime gaps is at most <math>16</math>.
* wThe work of Dan Goldston, J´anos Pintz, and Cem Yıldırım also shows that, unconditional to any conjecture, the limit inferior of the ratio of prime gap to the logarithm of the prime is zero.

===Average prime gap===

The [[prime number theorem]] states that the average prime gap is the natural logarithm of the prime.

===Supremum of prime gaps===

* [[Bertrand's postulate]] (which is in fact a theorem) states that there always exists a prime between any number and its double.