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Given some arbitrary prime number $p_n$

How many prime numbers can we prove exist which are smaller than ${p_n}^2?$

...Obviously we can say that $n$ primes must exist, but I think I can prove that $n + p_n - 1$ prime numbers must exists which are smaller than ${p_n}^2$.

Are there any better results?

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    $\begingroup$ The prime number theorem gives much better bounds. $\endgroup$ Nov 12, 2019 at 8:59
  • $\begingroup$ Isn't it an approximation? $\endgroup$ Nov 12, 2019 at 9:00
  • $\begingroup$ The link I included gives concrete inequalities. $\endgroup$ Nov 12, 2019 at 9:01
  • $\begingroup$ Wikipedia's banned in Turkey unfortunately... which concrete inequality is better? $\endgroup$ Nov 12, 2019 at 9:02
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    $\begingroup$ just by Bertrand's postulate and algebra you can guess $k^2+k$ primes very roughly. where $p_n=2\cdot k+1$ or log of it base 2 rather. $\endgroup$
    – user645636
    Nov 12, 2019 at 13:15

1 Answer 1

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The prime number theorem gives much better bounds. Here is a screenshot of some of those bounds, for those who can't use the link. The famous paper of Rosser and Schoenfeld is a good source for such inequalities.

screenshot from Wikipedia

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    $\begingroup$ Is there anyone who can see this site but can't see Wikipedia? $\endgroup$
    – lhf
    Nov 12, 2019 at 10:55
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    $\begingroup$ Turkey apparently $\endgroup$
    – user645636
    Nov 12, 2019 at 13:09

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