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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.

Joseph Martin
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    For any $k > 1$, there is at least 1 primes between $n$ and $kn$ for large enough $n$. This means that for all $N$, there are at least $N$ primes between $n$ and $2n$ for large enough $n$. (EG Set $k^N < 2$.) – Calvin Lin Mar 01 '23 at 01:41
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    @CalvinLin Thanks for your reply. However I'm interested in looking at a provable bound on $n$ for that to be true, and not simply "large enough". Again thanks for yours time. – Joseph Martin Mar 01 '23 at 02:41
  • How did you use a program to show something for (all) $n \ge 10$? – ronno Mar 22 '23 at 14:55

1 Answers1

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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

For any integer $n$ and $k$, if $n \ge 1.1\ln(2.5k)$, there are at least $k-1$ primes between $n$ and $kn$.

Both of these are relatively new results so I don't think much has been improved since their publication.

Nilotpal Sinha
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