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Since Zhang Yitang announced his proof that there are an infinite number of prime pairs with gaps less than 70 million, there's been a lot of work toward possibly proving the twin prime conjecture from his approach, or at least reducing the bound as much as possible - currently to less than 300,000. If we say that gaps below the current limit of this approach are "small" gaps, and gaps above that limit are "large" gaps, then my question is: What do we know about the number of prime pairs that differ by a given large gap?

Obviously, for odd gaps one member of the pair must be 2, and there will be at most one other prime that can result in a given gap. By the Prime Number Theorem, most specific large odd gaps will have exactly zero prime pairs that differ by that amount. But it's not clear to me what we know about large even gaps:

  1. For any specific large even gap, are there an infinite number of prime pairs with that exact gap?

  2. If we don't know this, can we prove that for any large even gap there are either an infinite number of prime pairs separated by that distance, or zero prime pairs separated by that distance?

  3. Can we at least say that there is some very large gap distance (that is, much larger than 70 million) above which condition 2 obtains?

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There is no direct way to turn the given result into a statement about infinitely repeated "large" gaps. The result only directly shows that at least one gap (small by your terminology) must repeat infinitely often. That can be seen by a simple pigeonhole argument. –  Thomas Andrews Jun 10 '13 at 16:22

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All of your questions are beyond current mathematical terminology. The answer to #1 is conjectured to be "yes". This is called Polignac's conjecture.

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This is helpful, thanks. Sometimes just finding out what to search for on Wikipedia is a big chore. –  Teve Torbes Jun 10 '13 at 17:59

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