Possible values for differences of primes 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.
 A: 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 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.  
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.
A: A survey on this topic can be found on http://www.ams.org/bull/2007-44-01/S0273-0979-06-01142-6/home.html . The distribution of primes is conjecturally described by the Cramer model, so you can find a desired conjectural evaluation of your density. Not so much can be proven however...
A: OK, I just saw this...a conjecture I made a few years ago covers this area
It relates d to OEIS A129912 entries, which are derived from the primorials.
Note that the d spoken to would only be a subset of those possible but these are guaranteed.  A quick example is the prime 189239, which is offset from A129912(24) by 9059.  So, the primes 189239 and 9059 have d=180180.
Note the "adjacency" part of the conjecture.  The conjecture reads as follows:
"Every prime number >2 must have an absolute distance to a sequence entry (primorials, 

primorial products) that is itself prime, aside from the special cases prime=2 and those primes immediately adjacent to a sequence entry (primorials, primorial products). 
The property is required but not sufficient ...it considers distances no larger than the candidate"
Rephrasing, this merely means every odd prime number must either be adjacent to, or a prime distance away from a primorial or primorial product. (the distance will be a prime smaller than the candidate)
Now obviously there are many other sets as you say ie 23-19,29-19,etc which lie outside the above. 
