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Polignac's conjecture states that for any positive even integer $K$, there exist infinitely many pairs of primes such that their difference is $K$.

I am interested the status in a much weaker form of the conjecture:

Is it true that for all even numbers $K$, there exist primes $p,q$ with $p-q=K$?

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  • $\begingroup$ This sounds like it should be so easy in the current formulation, but then when I rephrase it as "What is the state of the strong Goldbach conjecture except with the + changed to a -?" it sounds potentially much harder. $\endgroup$
    – Daniel McLaury
    Commented May 12, 2016 at 9:13

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I think this is open just as much as the original conjecture. In fact, in analytic number theory, we usually prove the existence of certain objects by showing that there are many of them (certainly infinitely many, which can be further defined by density etc.).

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