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.



Possible values for differences of primes
Since there are infinitely many primes, the set $K$ is certainly infinite, so in the expression $\frac{|K|}{|\mathbb{Z}^+|}$, you are attempting to divide two infinite cardinalities. This is not a meaningfully defined operation.


Not so much is known about the set $K$ unconditionally. However, an old conjecture of [Alphonse de Polignac](http://en.wikipedia.org/wiki/Alphonse_de_Polignac) states that for every positive integer $k$, there are infinitely many pairs of primes $p$ and $q$ such that $p-q =2k$. This is significantly stronger than saying that every even positive integer lies in $K$. de Polignac's conjecture is in turn a special case of a much broader conjecture which is, however, still widely believed to be true and often used by 21st century mathematicians to prove conditional results: [Schinzel's Hypothesis H](http://en.wikipedia.org/wiki/Schinzel%27s_hypothesis_H). 


On the other hand, $K$ can only contain an odd positive integer $n$ if $n+2$ is prime. The set of such numbers has density zero. So, assuming Schinzel / de Polignac, the set $K$ has asymptotic density $\frac{1}{2}$, i.e.,


$\lim\_{N \rightarrow \infty} \frac{|K \cap [1,N]|}{N} = \frac{1}{2}$. 


Perhaps an actual analytic number theorist can tell us how close we are to knowing this density result unconditionally.


