# Do arbitrary $K$-twin primes exist?

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 $p-q=K$?

• 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. – Daniel McLaury May 12 '16 at 9:13