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Let $q \geq 5$ be a prime number, and consider : $N_q = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}}$

Using Chinese remainder theorem we can show that :

$$\#\{(b,b+2) \in \mathbb{N}^2 \, | \, b+2 \leq N_q \text{ and } \gcd(b,N_q)=\gcd(b+2,N_q)=1\} = \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize (p-2)}}-1$$

Let $r$ be the next prime to $q$.

We have : $\gcd(n, N_q) = 1 \text{ and } q < n < r^2 \implies n \in \mathbb{P}$

Then if we prove that for every $q$ the region $q+1,q+2,\cdots,r^2-1$ has at least a tuple $(b,b+2)$ coprime to $N_q$ we prove the twin prime conjecture.

My question is : has this region been studied before ? and what are the most important results found about it?

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    $\begingroup$ As you see sieving the large region $[1,N_q]$ is not the same at all as the small one $[1,r^2]$. In TerryTao's blog on sieves, k-upples and AP problems, they show how it helps to shift $[1,r^2] \subset [1,N_q]$ to $[t+1,t+r^2]$ for smart values of $t$ $\endgroup$
    – reuns
    Commented Jul 31, 2019 at 0:46

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