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 $q+1,q+2,\cdots,r^2-1$ had at least a tuple $(b,b+2)$ coprime to $N_q$ we prove the **twin prime conjecture**.

**My question is :** this region is studied before ? and what is the most importante results founded about it?