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A proof I'm writing rests on something I can't prove, probably beyond my knowledge, but it seems right:

For any two primes $p_k, p_l$ (not necessarily consecutive) such that the distance between them $|p_l - p_k| = n$, there exist infinitely many other primes such that the distance between them is also $n$.

I can't figure out a way to show this; I'm guessing it's probably a known result and referring to it would be enough.

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    This is false if $p = 2$ :-) For odd primes, see http://en.wikipedia.org/wiki/Polignac%27s_conjecture2011-11-06
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    Maybe not false, but not yet proved? It is the twin-prime-conjecture.2015-01-26

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