# Prime gap

From Number

## Contents

- 1 Definition
- 2 Facts
- 2.1 Basic facts (lower bound on limit inferior, limit superior is infinity)
- 2.2 Conjectures and advanced facts on upper bounds on limit inferior and occurrence of small prime gaps
- 2.3 Conjectures and advanced facts on maximum prime gaps (growth of limit superior relative to size of numbers)
- 2.4 Other related facts

## Definition

The **prime gap** between a prime and its successor prime is the difference . In other words, a prime gap is a gap between two successive primes.

## Facts

We are interested in three broad things:

- How frequently does a given prime gap occur?
- The limit inferior of prime gaps, i.e., as we go to infinity, what is the limiting value of smallest prime gaps?
- The limit superior of prime gaps, i.e., as we go to infinity, what is the limiting value of largest prime gaps?

### Basic facts (lower bound on limit inferior, limit superior is infinity)

- A prime gap of occurs between and , and never again. All other prime gaps are even, and at least .
- There exist arbitrarily large prime gaps: This is because there exist arbitrarily large sequences of consecutive composite integer. For instance, for any , the sequence is a sequence of composite integers.

### Conjectures and advanced facts on upper bounds on limit inferior and occurrence of small prime gaps

Name of conjecture/fact | Statement | Status |
---|---|---|

twin prime conjecture | There exist arbitrarily large pairs of twin primes -- successive primes with a gap of two. Equivalently, the limit inferior of prime gaps is exactly 2. | open |

Polignac's conjecture | For any natural number , the prime gap occurs for arbitrarily large pairs of primes | open; stronger than twin primes conjecture |

Goldston-Pintz-Yildirim theorem on prime gaps conditional to Elliott-Halberstam | there exist infinitely many pairs of consecutive primes with prime gap at most 16 | proof conditional to Elliott-Halberstam conjecture |

Zhang's theorem on bounded prime gaps | There exist infinitely many pairs of primes that differ by at most 70 million. In other words, there eixsts a prime gap that is at most 70 million that occurs infinitely often. Subsequent work by the Polymath project brought the bound down to 4680 | Closed, see here and here. |

Maynard's theorem on bounded prime gaps | There exist infinitely many pairs of primes that differ by at most 600. Subsequent work brought the bound done to 270. | Closed, see here and here. |

### Conjectures and advanced facts on maximum prime gaps (growth of limit superior relative to size of numbers)

Name of conjecture/fact | Statement | Function (big-O) | Status |
---|---|---|---|

Cramér's prime gap conjecture | For any prime , the prime gap between and the next prime is at most , fixed | open | |

Prime-between-squares conjecture | There exists a prime between any two successive squares. Puts upper bound of on prime gap | open | |

(corollary of) Generalized Riemann hypothesis | The prime gap between a prime and the next prime is | open | |

exponent bound for prime gap of 0.535 | The prime gap between and the next prime is at most | proved | |

(corollary of) prime number theorem | there exists a prime between and for any , for large enough (dependent on ) | proved | |

Bertrand's postulate | there exists a prime between and | proved |