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Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$? Where $\pi(n)$ is the prime counting function. Using PMT it seems asymptotically $\ p_n^{\pi(n)} \leq x^n$ where $e \leq x$

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    Yes, but this statement is often used as an ingredient in the proof of PMT, so it would be nice to see a simple proof.2011-01-02

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Using $\pi(x) \le 1.25066 \frac{x}{\log x}$ for all $x>1$ (from Rosser and Schoenfeld), you have $(p_n)^{\pi(n)} \le e^{1.25066 n} < 3.5^n$ for all $n\ge 2$.

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    Just so it's clear for future reference, the paper by Rosser and Schoenfeld is [Approximate formulas for some functions of prime numbers](http://projecteuclid.org/euclid.ijm/1255631807), equation 3.6.2011-04-29
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Yes,

Asymptotically you have

$(p_n)^{\pi(n)} \leq n^{n/\log n} = e^n$

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    @arjang: Yes, for any c > 1, have $\pi(n) \le c\ n/\log n$ for sufficiently large $n$.2011-01-02