This Wikipedia article shows us what are the first twin primes. Here we state the notation for each twin prime pair writing $(q_n,2+q_n)$ for $n\geq 1$ (for example $q_1=3$ and $q_8=71$).

On the other hand this Wikipedia show what is the Bonse's inequality.

Question. I would like to know if, on assumption of some form of the Twin Prime conjecture, you can set a conjecture of the same kind of Bonse's inequality when one writes the elements of the sequence $(q_n)_{n\geq 1}$ instead of the full sequence of prime numbers in Bonse's inequality. See below my thoughts in next examples. Many thanks.

Example 1. One has that $$59^2<q_1q_2q_3q_4q_5q_6=3335145,$$ thus the similar inequality (than Bonse's inequality that one writes for twin primes) holds for $n=4$.

Example 2. Maybe is it possible to get a more sharper inequality, and with mathematical meaning, if one can state that there exists a $N$ and positive rational numbers $a,b$ such that $$(q_{n+1})^a<\left(\prod_{k=1}^n q_k\right)^b$$ holds for all $n>N$, on assumption that there are infinitely many twin primes.

That is: can you write, on assumption of a twin prime conjecture, a similar inequality of the same kind of Bonse's inequality now for twin primes?

In the Wikipedia's article is the reference to genuine Bonse's inequality.

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    $\begingroup$ Do you realize this is unreadable ? And the twin prime conjecture can be true with gigantic gaps between consecutive pairs of twin primes. $\endgroup$ – reuns Jun 13 '17 at 11:45
  • $\begingroup$ Thanks for your attention. Yes I am agree that maybe it is possible improve my english here. If some user want to fix some about the grammar it is the best. About your comment I don't know how is related with my question, now. Merci @user1952009 $\endgroup$ – user243301 Jun 13 '17 at 11:48
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    $\begingroup$ The Bonse's inequality is a bound for $p_{n+1}$, and hence for the gap between $p_n$ and $p_{n+1}$... $\endgroup$ – reuns Jun 13 '17 at 13:22
  • $\begingroup$ Feel free to add your reasonings and mathematics as an answer, if is feasible to state a conjecture in the way that I've evoked: a Bonse's inequality for twin primes. Many thanks @user1952009 $\endgroup$ – user243301 Jun 13 '17 at 13:50
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    $\begingroup$ What are you talking about ? What I wrote is trivial. Yes you can make all the conjectures you'd like about twin primes. But how does it help in anything ? The main conjecture about prime numbers is the probabilistic model, which implies the RH, the twin prime conjecture and an asymptotic distribution of twin primes. $\endgroup$ – reuns Jun 13 '17 at 14:00

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