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 2 years ago
Let \( \pi(n)\) be the number of primes less or equal to n.
Show that
\[
n^{\pi(2n)\pi(n)}<4^{n}
\]
 2 years ago
Let \( \pi(n)\) be the number of primes less or equal to n. Show that \[ n^{\pi(2n)\pi(n)}<4^{n} \]

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mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2sir i tried induction..but no result are we supposed to prove by induction or not?

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.0The proof I know does not need induction.

sauravshakya
 2 years ago
Best ResponseYou've already chosen the best response.0@mukushla got any clue?

eliassaab
 2 years ago
Best ResponseYou've already chosen the best response.0Hint \[ 4^n=(1+1)^{2n}> {2n \choose n} \]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.2for \(n\le p_k\le 2n\)\[\prod p_k  \frac{(n+1)(n+2)...(2n)}{n!}={2n \choose n} \Rightarrow\prod p_k <{2n \choose n}\]\[n^{\pi(2n)\pi(n)}< \prod p_k<\binom{2n}{n}< 2^{2n}=4^{n}\]
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