# Prime Numbers in this region

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?

• 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 $t$ – reuns Jul 31 '19 at 0:46
• @reuns, can you please specify the article in TerryTao's blog ? – LAGRIDA Nov 26 '19 at 12:28