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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 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===
-* [[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.