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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.

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3 Answers

up vote 2 down vote accepted

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.

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Thanks, Pete. I meant the asymptotic density, not the actual value of the expression. For odd $d$, does anyone else have any references about the types of numbers where $d+2$ is prime? –  simpavid Mar 29 '10 at 7:41
    
Here You may find another problem related to Polingac and prime gap problem: it is called Andrica's conjecture en.wikipedia.org/wiki/Andrica%27s_conjecture . It is worth of note because of large empirical evidence. –  kakaz Mar 29 '10 at 8:06
    
@simpavid: at the serious risk of pointing out the obvious, it seems that the simplest and best way to view the set of numbers in your comment is as the set of prime numbers translated by $-2$. –  Pete L. Clark Mar 29 '10 at 13:32
    
@Pete L. Clark: I guess the question is then, how many members of the set of prime numbers translated by $-2$ are also prime? Thanks for the help. –  simpavid Mar 29 '10 at 22:11
    
Brun proved (a bit before 1920) that "almost no" primes translated by -2 are also prime. More explicitly, we now know the following statement (Brun's original statement was slightly weaker): while the number of primes up to x is asymptotic to x/log x, the number of primes p up to x such that p-2 is also prime is bounded by some constant times $x/(\log x)^2$. –  Greg Martin Apr 3 '10 at 6:24
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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...

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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.

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