Prime gap

Definition

The prime gap between a prime $p$ and its successor prime $q$ is the difference $q-p$. 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 $1$ occurs between $2$ and $3$, and never again. All other prime gaps are even, and at least $2$.
• There exist arbitrarily large prime gaps: This is because there exist arbitrarily large sequences of consecutive composite integer. For instance, for any $n>1$, the sequence $n!+2,n!+3,\dots ,n!+n$ 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 $n$, the prime gap $2n$ 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 $m$ 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 $p$, the prime gap between $p$ and the next prime is at most $c(\log p)^{2}$, $c$ fixed	$O((\log p)^{2})$	open
Prime-between-squares conjecture	There exists a prime between any two successive squares. Puts upper bound of $1+2{\sqrt {p}}$ on prime gap	$O({\sqrt {p}})$	open
(corollary of) Generalized Riemann hypothesis	The prime gap between a prime $p$ and the next prime is $O({\sqrt {p}}(\log p))$	$O({\sqrt {p}}\log p)$	open
exponent bound for prime gap of 0.535	The prime gap between $p$ and the next prime is at most $p^{0.535}$	$O(p^{0.535})$	proved
(corollary of) prime number theorem	there exists a prime between $n$ and $\alpha n$ for any $\alpha >1$, for $n$ large enough (dependent on $\alpha$)	$O(n)$	proved
Bertrand's postulate	there exists a prime between $n$ and $2n$	$O(n)$	proved

Other related facts

• Chen's theorem on primes and semiprimes with fixed separation