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
< x, where p_n
is the nth 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)



The short answer is no, though if one assumes the ElliotHalberstam 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. 


Assuming this is not wrong, the short answer is now 70 million. See the current front page of the AMS ...



The limit you mention isn't welldefined, but you can instead take the lim sup. An elementary argument shows that there's no such x that upper bounds prime gaps; if there were, then (x+2)! + 2, (x+2)! + 3, ..., (x+2)! + x+1, (x+2)! + x+2 are all composite, which would lead to a contradiction. You can also ask "What's the smallest x such that p {n+1}  p n < x infinitely often?," which is probably closer to what you intended. I don't think that any such constant x is known to exist unconditionally, but assuming a strong conjecture known as the ElliottHalberstam conjecture, Goldston, Pintz, and Yildirim have shown that you can take x = 20. 

