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

Basic facts

• 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 minimum prime gaps

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

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