# Twin prime conjecture

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This article states a conjecture about there existing infinitely many of the following numbers/structures: twin primes

View other infinitude conjectures | View infinitude facts

## Contents

## Statement

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

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

## 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 . 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.