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The conjectural density of twin primes is $\frac {c\cdot n}{(\log n)^2}$ at a $c>0$.

Consider integers of form $p,p+1=2^tq,p+2=r$ where $p,q,r$ are primes and $t\geq1$ holds.

  1. Is there any reason to believe there are infinite of them at a given $t\geq1$? Is there a conjectural density for such triples at a given $t$?

  2. Is there any reason to believe there are infinite of them with $t$ not fixed? Is there a conjectural density for such triples with $t$ not fixed?

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Your questions (more precisely their affirmative answers) are special cases of the generalized Hardy-Littlewood conjecture. You can read about this conjecture in Linear equations in primes. See especially Conjecture 1.4 on Page 5 and the subsequent remarks on Page 6.

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  • $\begingroup$ interesting.... $\endgroup$
    – Turbo
    Commented Nov 15, 2017 at 0:02
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    $\begingroup$ @Turbo: Thank you. I think your "accept" was the quickest one for me on MathOverflow so far. Well under a minute! $\endgroup$
    – GH from MO
    Commented Nov 15, 2017 at 0:03
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    $\begingroup$ @GHfromMO I believe it is relevant with his nickname ^^ $\endgroup$
    – Desiderius Severus
    Commented Nov 15, 2017 at 7:11

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