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How can one find the limit as M approaches infinity of the ratio of the number of primes p to the number of primes q all less then M.

Where every p satisfy: p+42 is prime, and p+20 is prime.
And every q satisfy: q+2 is prime and q+18 is prime and q+44 is prime.

Which seems to converge to around 10.

How can one find the general case for this ratio?

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I don't know if I understood your question correctly, but the distribution of primes is described by the prime number theorem. –  Fernando Martin Oct 1 '11 at 20:17
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Even though it is reasonable to suppose that there are infinitely many $p$, $q$ satisfying your conditions, I am reasonably sure that this has not been proved, and is well beyond current techniques. One can only make plausible guesses about the long term behaviour of the ratio, based on reasonable but unproved conjectures that apart from obvious congruential conditions, primes behave roughly randomly. –  André Nicolas Oct 1 '11 at 20:29
    
@Andr What are good (introductory) books or papers to everything we know about the prime numbers and their distribution? –  user1708 Oct 1 '11 at 20:38
    
As somebody said, prime numbers are the mathematical object such that anybody can do some statement about, that will turn out to be a very hard mathematical conjecture. –  natema Oct 1 '11 at 21:09
    
Relevant: Frequencies of successive pairs of prime residues –  anon Oct 1 '11 at 21:29
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2 Answers

up vote 4 down vote accepted

If you assume the Hardy-Littlewood k-tuples conjecture (warning: PDF file), the answer is $M = \infty$, because

  1. neither of the sets $P = \{0,20,42\}$ and $Q = \{0,2,18,44\}$ is a complete residue system for any prime; and
  2. $|P| < |Q|$.

If you don't assume the Hardy-Littlewood k-tuples conjecture, then I reckon this is an unsolved problem.

BTW: The reason your computation appears to converge to a limit is just that your computer can't count to $\infty$.

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This should be equivalent to $p'$ and $p'+22$ are primes, and $q',q'+16,q'+42$ are primes.

Note, that figuring out distribution of $p,p++2$ both primes is the Twin prime conjecture, http://en.wikipedia.org/wiki/Twin_prime so there is no luck on even figuring out if there are infinitely many primes $p$ satisfying that $p+22$ is also a prime number.

Hence, your problem is super hard, and solving it would probably imply that you will also be able to solve this conjecture: http://en.wikipedia.org/wiki/Polignac%27s_conjecture

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Just a nitpick: the question specified that $p$ and $q$ are also prime. But you're right about super-hard. –  TonyK Oct 1 '11 at 21:27
    
Oh, definition spanned over several paragraphs, did not realize that... –  Paxinum Oct 2 '11 at 8:00
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