'Bounded gaps between primes':

"It is proved that

[tex]\lim inf_{n \to \infty}(p_{n+1 } - p_{n } ) < 7 * 10^{7}[/tex],

where [tex]p_{n}[/tex] is the n-th prime.

Our method is a refinement of the recent work of Goldston, Pintz and Yildirim on the small

gaps between consecutive primes. A major ingredient of the proof is a stronger version of the

Bombieri-Vinogradov theorem that is applicable when the moduli are free from large prime

divisors only (see Theorem 2 below), but it is adequate for our purpose..." -- Yitang Zhang,

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.308.998&rep=rep1&type=pdf