Twin Primes between $n$ and $2n$

Question

Is it theoretically possible for there to always be a twin prime pair between $n$ and $2n$ for all sufficiently large $n$ (assuming of course that there are infinitely many twin primes) or would this contradict Brun's Theorem? Thanks.

Answer 1

Brun's theorem follows from the bound $$ \pi_2(x) = O\left(\frac{x(\log\log x)^2}{(\log x)^2}\right), $$ where $\pi_2(x)$ is the number of twin primes less than $x$. On the other hand, the existence of twin primes between each $n$ and $2n$ only implies $\pi_2(x) = \Omega(\log x)$ (as Myself notes in the comments), so there is no contradiction.

Moreover, it is conjectured that $\pi_2(x) \sim 2C_2 \displaystyle\frac{x}{(\log x)^2}$ for some constant $C_2$. This implies that for large enough $n$, there exist many twin primes between each $n$ and $2n$ (in fact, roughly $C_2 \displaystyle\frac{n}{(\log n)^2}$).