Bounded gaps between primes occur when there is a finite number 
such that infinitely many pairs of consecutive prime
numbers differ by at most 
. If 
denotes the 
th prime, define the limit inferior
 |
The assertion that bounded gaps between primes exist is the statement 
.
Zhang (2014) first proved this unconditionally, obtaining 
. In 2013, the Polymath8 project reduced the
bound to 
(Polymath 2014a), Maynard (2015) reduced it to 
, and further work of the Polymath8 project established
the unconditional bound 
(Polymath 2014ab). Maynard also proved that 
is finite for every positive integer 
(Maynard 2015). Consequently, for every fixed 
, infinitely many intervals of some fixed length contain at
least 
primes.
Stronger conditional bounds are known. Assuming the Elliott-Halberstam conjecture, Maynard (2015) proved 
and 
. The Polymath8 project improved the latter bound
to 
under the same conjecture. Under the stronger generalized form of the Elliott-Halberstam
conjecture, it proved 
and 
(Polymath 2014b). These are conditional bounds,
unlike the unconditional bound 
.
Except for the gap following 2, every prime gap is even. Therefore, 
implies that at least one even gap among 2, 4, ...,
246 occurs infinitely often between consecutive primes. It does not specify which
one. In particular, none of the unconditional or conditional bounds proves the twin prime conjecture, which is equivalent to
.
