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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](http://en.wikipedia.org/wiki/Prime_number_theorem).2011-10-01
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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.2011-10-01
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    @Andr What are good (introductory) books or papers to everything we know about the prime numbers and their distribution?2011-10-01
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    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.2011-10-01
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    Relevant: *[Frequencies of successive pairs of prime residues](https://www2.bc.edu/~ashav/Papers/ABGS-PrimePairsFinal.pdf)*2011-10-01

2 Answers 2

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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.2011-10-01
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    Oh, definition spanned over several paragraphs, did not realize that...2011-10-02