It is well known that there is a prime number between $n$ and $2n$ for all $n$. I decided to go deeper: is there a lower bound on the number of primes between $n$ and $2n$ for "large enough" $n$?
For instance, I found empirically that $\pi(n)-\pi(n/2)\ge \sqrt n$ at around $n\ge 100$ (In particular, I also used a program to show $\pi(n^2)-\pi(n^2/2)\ge n$ for $n\ge 10$). Does there exist a nice elementary proof of this?
Obviously, we can show this result asymptotically for very great $n$. I ask to consider an elementary proof of the above bound.
From Theorem 1.2 and 1.3 of Axlar's 2017 paper, we have for $x \ge 49$,
$$ \pi(x) < \frac{x}{\ln x - 1 - \frac{1}{\ln x} - \frac{3.15}{\ln^2 x} - \frac{12.85}{\ln^3 x} - \frac{71.3}{\ln^4 x} - \frac{463.2275}{\ln^5 x} - \frac{4584}{\ln^6 x} } $$
and for $x \ge 19,033,744,403$, $$ \pi(x) > \frac{x}{\ln x - 1 - \frac{1}{\ln x} - \frac{2.85}{\ln^2 x} - \frac{13.15}{\ln^3 x} - \frac{70.7}{\ln^4 x} - \frac{458.7275}{\ln^5 x} - \frac{3428.7225}{\ln^6 x} } $$
Plug in these inequalities you will get the best currently known bounds on $\pi(n) - \pi(n/2)$.
Another way to look at a stronger form of Bertrand's postulate is found in Theorem 3.6 of Das and Paul's 2019 paper where they proved that
Both of these are relatively new results so I don't think much has been improved since their publication.