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Proposition: $\pi (2n) \ge \dfrac{\log \binom{ 2n }{n} }{\log 2n}$

Since this is a follow up proposition to this one: How can we show that $\operatorname{ord}_{p}\left(\binom{2n}n\right) \le \frac{\log 2n}{\log p}$

We can use : $\operatorname{ord}_{p}\left(\binom{2n}n\right) \le \frac{\log 2n}{\log p}$

attempt :

$n \log 2 \le \sum_{p\le 2n}\left[\frac{\log 2n}{\log p}\right]\log p \le \sum_{p \le 2n} \log 2n = \pi(2n) \log 2n$

(don't see how to "finish", if this is correct at all... )

  • 1
    What does $n\log 2$ have to do with $\log\binom{2n}{n}$?2012-03-15

1 Answers 1

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Hint:

$\rm p\Big|\binom{2n}{n}\implies p|(2n)! \implies p\le 2n; $

$\rm m=\prod_{p|n} p^{\operatorname{ord}_p(m)}: \quad m=\binom{2n}{n}\implies \log \binom{2n}{n}=\sum_{p\le 2n}(\log p)\cdot\operatorname{ord}_p\binom{2n}{n} $

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    Shouldn't that be $p\mid m$ in the product limits?2018-09-19