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 $p-q=K$?
> 
> 
> 



Do arbitrary $K$-twin primes exist?
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.).


