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 ![]() ![]() |
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 ![]() |
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 ![]() ![]() ![]() ![]() |
![]() |
open |
Prime-between-squares conjecture | There exists a prime between any two successive squares. Puts upper bound of ![]() |
![]() |
open |
(corollary of) Generalized Riemann hypothesis | The prime gap between a prime ![]() ![]() |
![]() |
open |
exponent bound for prime gap of 0.535 | The prime gap between ![]() ![]() |
![]() |
proved |
(corollary of) prime number theorem | there exists a prime between ![]() ![]() ![]() ![]() ![]() |
![]() |
proved |
Bertrand's postulate | there exists a prime between ![]() ![]() |
![]() |
proved |