If $P, Q$ are prime, and $P > Q$, then let $K$ be the set of all numbers $(P-Q)$. Is there a way to determine $\frac{|K|}{|\mathbb{Z}^+|}$? Is this even a converging value? What kind of numbers are in set $K$?

So far: if $P-Q = d$ is odd, then $P, Q$ are of different parity and $Q = 2$, so $d = P-2$.

But, if $d$ is even, then $P, Q$ are both odd, which means finding primes that are $d$ away from each other, where $d$ is an even number. For how many values of $d$ is this possible?

Though this seems similar to the twin primes conjecture, note that here we only ask if a value of $d$ is possible, not how many such pairs there are.

Sorry if this is in fact a trivial problem, I'm not very experienced in mathematics.