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If $p_n$ is the nth prime number does $\frac{p_{n+1}}{p_n}$ approach any particular number when $n$ is large? What about $\frac{p_n}{p_{n+1}}$? Is it possible to prove that the limits converge or diverge with elementary methods?

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marked as duplicate by mfl, Dominik, Community Nov 30 '16 at 14:49

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    $\begingroup$ $p_n \sim n\ln n$ by PNT, hence $1$ for both questions. $\endgroup$ – Paolo Leonetti Nov 30 '16 at 14:44
  • $\begingroup$ @PaoloLeonetti Interesting, thank you. $\endgroup$ – David Nov 30 '16 at 14:46
  • $\begingroup$ @mfl I'll check it out, thanks. $\endgroup$ – David Nov 30 '16 at 14:47
  • $\begingroup$ Although I don't know if the prime number theorem falls under "elementary methods". $\endgroup$ – David Nov 30 '16 at 14:49