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 two broad things:

• 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

Name of conjecture/fact	Statement	Status
twin primes conjecture	there exist arbitrarily large pairs of twin primes -- successive primes with a gap of two.	open

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