I just want make thing clear for myself. Others may have asked before in different ways. Does Yitang Zhang's paper prove that for any given length gap $g_n > N$ there is a prime $p_n$ for which there are infinitely many prime numbers $p_k > p_n$ which have $g_k = g_n$? Or, is it something else? If I do have this correctly, is p_n equal to the first prime of gap $N$? Does this mean that for any gap greater than g_n there is no "last one" only "last know use"?
closed as offtopic by Anthony Quas, Ryan Budney, Willie Wong, Lucia, Andrey Rekalo Apr 10 '14 at 17:45This question appears to be offtopic. The users who voted to close gave this specific reason:



Yitang Zhang's paper proves that there is a gap less than or equal to 70 million that occurs infinitely often. More precisely, the paper proves that if you take 3.5 million integers that do not form a complete residue system modulo any integer greater than 1, then one of the pairwise distances among these integers occurs infinitely often as the difference of two distinct primes. We do not know a single value, written down as a concrete number like 2014, that provably occurs as a difference of two distinct primes infinitely often. P.S. Due to the efforts of Maynard, Pintz, Tao and other members of the PolyMath8 group, the 70 million above has been improved to 252, and the 3.5 million above has been improved to 51. Added. The current record is now 50 for the size of the tuple in the method, yielding the bound 246 for a gap size that occurs infinitely often among the primes. The relevant papers by the Polymath8 group have appeared: New equidistribution estimates of Zhang type, Algebra & Number Theory 8 (2014), 20672199. Variants of the Selberg sieve, and bounded intervals containing many primes, Res. Math. Sci. 1 (2014), no. 12, 83 pp. 


Zhang's paper proves that there are infinitely many prime pairs whose difference is less than 70,000,000 (this has since been reduced). What that means is that there is some $k \in \{2, 4, 6, \dots, 70,000,000\}$ such that, for infinitely many $n$, $p_{n+1}  p_n = k$. It doesn't tell us where these pairs are and it doesn't tell us whether there are any $k_0 > k$ such that there are infinitely many prime pairs with gap $k_0$ (although Pintz has proved that this does, in fact, happen (see the paper "On the difference of primes" by Janos Pintz)) I hope this answers the question. 

