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I have been reading a bit about Zhang's proof and the associated Polymath8 project. Though Tao's high level summary http://terrytao.wordpress.com/2013/06/30/bounded-gaps-between-primes-polymath8-a-progress-report/ is interesting it still is rather technical. So without any knowledge of the techniques I am unable to get an intuition for the following related problems.

Let $T_k$ be the assertion that there are infinitely many prime pairs with distance $k$. It is conjectured that this is true for any even $k$ and Zhang's theorem obviously implies that this is true for some $k$. That being said, there is no known $k$ for which this is true.

  1. Is there any hope that the method employed in the proof is able to show this for some explicit $k$, even if this $k$ is much larger than the bounds given so far. My inituition says no.

  2. The goal of Polymath8 is to lower the bound given in Zhang's paper as far as possible. On the other hand it is believed that $T_k$ holds for any even $k$. Is there any hope that the methods can also be adapted to show the following: For any $n$ there is some $k \geq n$ such that $T_k$ is true. This seems much more in the style of the original theorem than question 1, so here I am more optimistic, but I guess some deeper knowledge is required here.

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up vote 8 down vote accepted

Zhang's strategy shows that any admissible tuple of size h contains at least 2 primes infinity often, as long as h is larger than some threshold (the primary theoretical thrust of the polymath project has been to reduce the required value of h). This approach seems unable to give (1), but immediately gives (2).

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