Let $n, k$ are integers number such that $1<n \le k$, does always exist a prime number between $kn$ and $k(n+1)$?
When $n=1, k>1$ always exist a prime number between $k$ and $2k$ the question was proved Bertrand's postulate
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3$\begingroup$ If $k = n$ the question is whether there always exists a prime between $n^2$ and $n(n + 1)$. This is a strengthening of Legendre's conjecture, which is an open problem. $\endgroup$– RandomCommented Jun 15, 2021 at 10:53
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$\begingroup$ I don't know who vote to close, why? $\endgroup$– Đào Thanh OaiCommented Jun 15, 2021 at 10:59
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1$\begingroup$ I voted to close, because the question is a variant of Legendre's conjecture (which is probably true but out of reach). It is advisable to do some background reading before asking questions. A starting point here could be en.wikipedia.org/wiki/Cram%C3%A9r%27s_conjecture $\endgroup$– GH from MOCommented Jun 15, 2021 at 11:32
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$\begingroup$ The conjecture is not new, see Remark 2.7 of my 2015 paper available from maths.nju.edu.cn/~zwsun/160p.pdf . $\endgroup$– Zhi-Wei SunCommented Jun 15, 2021 at 12:17
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$\begingroup$ @GHfromMO Please, could you review help me? mathoverflow.net/questions/395913 $\endgroup$– Đào Thanh OaiCommented Jun 22, 2021 at 10:16
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1 Answer
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First of all, as mentioned by Random above, this is a very strong conjecture, because it is stronger than Legendre's conjecture. As far as I know, it is not known even if we assume the truth of Riemann Hypothesis and also some reasonable conjectures on distribution of imaginary parts of zeros, such as the Montgomery's pair correlation conjecture. However, it is known that $$ \pi(x+x^{0.525})-\pi(x)>0 $$ for large $x$. This means, for instance, that if $k$ is large and $$ n\leq k^{19/21}=k^{0.9047\ldots} $$ the conjecture is, in fact, true.
Also, there are results on the measure of large gaps. For example, $$ \sum_{\substack{p_n\leq x \\ p_{n+1}-p_n>\sqrt{p_n}}}(p_{n+1}-p_n)\ll x^{3/4+o(1)}, $$ see "The Riemann zeta-function: theory and applications" by A. Ivić, Theorem 12.14. This result implies that your conjecture is true for all $n$, except at most $k^{1/2+o(1)}$ for $k\to +\infty$.
answered Jun 15, 2021 at 11:16