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In my intuitive random search for conjectures I found $$n\neq 4\implies p_{n^2}\leq p_n^2<p_{4n^2}$$ It's tested for $n<1000$. I've looked at it the point of view of PNT but haven't the skills to prove or disprove it. Could there be counter examples out of reach of my conjecture seeking routines?

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    $\begingroup$ $p_n$ is roughly $n\log n$, so $p_n^2$ is roughly $n^2(\log n)^2$, and $p_{4n^2}$ is roughly $4n^2(\log4+2\log n)$, which should be smaller for all sufficiently large $n$. $\endgroup$ – Gerry Myerson Jul 24 '19 at 10:21
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This follows trivially from PNT.

For the left inequality. Since $p_n \sim n \log n$ hence $p_{n^2} \sim 2n^2 \log n$ but $p_n^2 \sim n^2 \log^2 n$ hence for some $n > n_0$, the growth rate of $p_n^2$ will exceed that of $p_{n^2}$.

For the right inequality, $p_n^2 \sim n^2 \log^2 n$ but $p_{4n^2} \sim 4n^2 \log 4n^2 = 4n^2(2\log 2 + 2\log n)$ so clearly this inequality will eventually fail.

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    $\begingroup$ You can do the other half of the question using the same logic. $\endgroup$ – Gerry Myerson Jul 24 '19 at 10:22
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    $\begingroup$ The PNT only shows that the proposer's conjecture is true for all sufficiently large $n$, but not necessarily for all $n>4.$ However, perhaps some cruder upper & lower bounds for $p_n$ that hold for all $n>4$ might suffice for this Q. $\endgroup$ – DanielWainfleet Jul 24 '19 at 12:16
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    $\begingroup$ @DanielWainfleet More precise values of the minimum point of validity can be found using Dusart's sharper upper and lower bounds on $p_n$. unilim.fr/laco/rapports/1998/R1998_06.pdf $\endgroup$ – NiloS Jul 24 '19 at 12:18

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