Is there known to be an $x$ such that for all positive integers $N$ there exists some $n>N$ such that $p\_{n+1}-p\_n \leq x$, where $p\_n$ is the $n$th prime? Or, in other words, is it known that limit as $n$ goes infinity of $p\_{n+1}-p\_n$ is not infinity? If such an $x$ is known to exist, what is the current best known $x$? (Showing $x=2$ would imply the Twin Prime Conjecture, of course.)



Is there a known bound on prime gaps?
(**Edit**: things have happened since the original post, changing the short answer to yes. See for example <http://arxiv.org/abs/1410.8400> for the status in 2014 where $x \leq 600$ unconditionally. GRP **End Edit**)


The short answer is no, though if one assumes the Elliot-Halberstam conjecture then one can take x=16. See


<http://arxiv.org/abs/math/0605696>


for a comprehensive survey of the best known results (both conditional and unconditional). 


There is also the Wikipedia article at


<http://en.wikipedia.org/wiki/Prime_gap>


although this is less comprehensive.


