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Let's define the notion of "central number" as follows: $n$ is said to be a central number if and only if there exist two consecutive odd primes $p$ and $q$ such that $n=(p+q)/2$. It is quite easy to check that the number of central numbers less than $x$ is asymptotically equal to $\pi(x)$. Let's now say that a central number $n$ is of type $k$ if and only if the least prime factor of $n$ equals $p_k$.

I have several questions:
1) Letting $N_k(x)$ be the number of central numbers of type $k$ less than $x$, is it true that $\displaystyle{\lim_{x\to\infty}\dfrac{N_{k}(x)}{\pi(x)}}$ exists and is positive for all positive $k$?
2) Does de Polignac's conjecture imply that there exists infinitely many central numbers of type $k$ for all positive $k$?
3) Does an affirmative answer to question 1) "conversely" imply de Polignac's conjecture?
Thanks in advance.

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I think one can imagine that the central numbers form a special enough subset that the range of types can be any superset of what is already observed, and that any nonempty subset of that superset can be the set of types for which there are infinitely many numbers of that type. Also, there is nothing besides hope to suggest that any of the desired limits exist. Further I do not hold any hope for an implication to exist between 2 !and 3 and de Polignac. Gerhard "The Concepts Might Hold Interest" Paseman, 2012.06.15 –  Gerhard Paseman Jun 15 '12 at 17:25
Also, you might observe that "most" of the observed central numbers are multiples of 2 or 3, and that only when you get to consecutive primes with a gap a multiple of 12 is there a chance for any other type to appear. The best you could hope for with current understanding is to resolve whether there are infinitely or finitely many central numbers outside of types 1 and 2, and I doubt that will be known anytime soon. Gerhard "Ask Me About Primorial Totients" Paseman, 2012.06.15 –  Gerhard Paseman Jun 15 '12 at 17:44
It follows from a result of Shiu (2000) "Strings of congruent primes" that there are arbitrarily long runs of consecutive primes congruent to a mod q. Taking q to have many small prime factors, this gives infinitely many central numbers of type >k for any k. Can this be sharpened to give exact type k? –  Erick Wong Jun 15 '12 at 20:25
I don't think one is allowed to choose q in that result, as otherwise several conjectures in prime gaps would be resolved. Can you say more about this result? Also, note that q has to be a multiple of 12 to aid in this problem. Gerhard "Ask Me About System Design" Paseman, 2012.06.15 –  Gerhard Paseman Jun 15 '12 at 20:39
Looking more closly at a 2010 paper (of friedman?) which builds on Shiu's result in short intervals, I agree with Erick's conclusion but not with his unspoken hope about sharpening. To me it feels too close to saying how very small the gaps can be, which would shed light on de Polignac. I also retract my above comment on conjecture resolution. Gerhard "Ask Me About Reading Twice" Paseman, 2012.06.15 –  Gerhard Paseman Jun 15 '12 at 21:06
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