Twin prime conjecture: Difference between revisions

Latest revision as of 21:10, 29 January 2014

This article states a conjecture about there existing infinitely many of the following numbers/structures: twin primes
View other infinitude conjectures | View infinitude facts

Statement

There are infinitely many twin primes. In other words, there are infinitely many odd primes $p$ such that $p+2$ is also a prime.

In other words, the limit inferior of all prime gaps is $2$ .

Relation with other conjectures and known facts

Limit inferior of prime gaps

The twin primes conjecture can be viewed as saying that the lim inf of prime gaps is $2$ . 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.

Generalizations

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.

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 {{prime gap conjecture}}
+{{infinitude conjecture|twin primes}}
 ==Statement==
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 ==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.
-* 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>.
+===Generalizations===
-* 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===
+* [[Polignac's conjecture]] states that every even number occurs infinitely often as a prime gap.
+* [[Dickson's conjecture]] generalizes the twin prime conjecture somewhat.
-The [[prime number theorem]] states that the average prime gap is the natural logarithm of the prime.
+* [[Schinzel's hypothesis H]] generalizes Dickson's conjecture.
+* [[Bateman-Horn conjecture]] generalizes Schinzel's Hypothesis H.
-===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.