Question

Let \( \pi(n)\) be the number of primes less or equal to n.

Show that

\[
n^{\pi(2n)-\pi(n)}<4^{n}

\]

Answer 1

sir i tried induction..but no result

are we supposed to prove by induction or not?

Answer 2

The proof I know does not need induction.

Answer 3

@mukushla got any clue?

Answer 4

nope :(

Answer 5

Hint
\[
4^n=(1+1)^{2n}> {2n \choose n}
\]

Answer 6

for \(n\le p_k\le 2n\)\[\prod p_k | \frac{(n+1)(n+2)...(2n)}{n!}={2n \choose n} \Rightarrow\prod p_k <{2n \choose n}\]\[n^{\pi(2n)-\pi(n)}< \prod p_k<\binom{2n}{n}< 2^{2n}=4^{n}\]