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We say that a $k$-tuple of integers $b_1, \cdots, b_k$ is admissible if for every prime $p$ there exists an integer $X$ such that none of the numbers $X + b_1, \cdots, X + b_k$ is divisible by $p$. Then the prime $k$-tuple conjecture can be stated as follows:

If $b_1, \cdots, b_k$ is an admissible $k$-tuple, then there exists infinitely many values of $X$ such that $X+b_1, \cdots, X+b_k$ are all prime.

Richards and Hensley has shown in 1974 that this conjecture is incompatible with another conjecture due to Hardy and Littlewood, which asserts that $\pi(M+N) \leq \pi(M) + \pi(N)$ for integers $M,N > 1$.

So my question is has either of these conjectures been confirmed to be true or false? Obviously if one of them is true then the other is automatically false. If a counter example exists for either problem, can anyone point out the counter example?

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It is currently believed that the second conjecture is likely false, but it hasn't been proven quite yet. There is an interval of size 3159 which is not prevented from having more primes than the initial segment of 3159 integers, which is how the first Hardy-Littlewood conjecture would refute the second one. See this wiki article for more information.

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  • $\begingroup$ But isn't this "show[ing] that there are ranges of integers where more primes can fit in those intervals than can fit into the initial segment of the integers" precisely the known connection pointed out in the question. $\endgroup$
    – user9072
    Commented Feb 3, 2012 at 20:05
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    $\begingroup$ Yes. And people currently strongly believe the first conjecture. There is all sorts of evidence towards it, but there doesn't seem to be any reason to believe the second conjecture. $\endgroup$
    – Pace Nielsen
    Commented Feb 3, 2012 at 20:07
  • $\begingroup$ There is current numerical work trying to show that 3159 is the smallest size where this happens: opertech.com/primes/k-tuples.html $\endgroup$
    – Pace Nielsen
    Commented Feb 3, 2012 at 20:09
  • $\begingroup$ Sorry for the nitpicking, but then why do you answer so indirectly and not just or at least also what you commented. After all the question is symmetric regarding the two conjectures. $\endgroup$
    – user9072
    Commented Feb 3, 2012 at 20:15
  • $\begingroup$ Brevity. After your comment I did try to expand my answer a little. Sorry I was so brief. I perhaps should have mentioned all of the results which tend to validate the first Hardy-Littlewood conjecture (such as the existence of infinitely many almost prime pairs). $\endgroup$
    – Pace Nielsen
    Commented Feb 6, 2012 at 16:58

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