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While reading this post, I stumbled across these definitions (Wiki_german)

$e = \lim_{n \to \infty} \sqrt[n]{n\#}$

and

$e = \lim_{n \to \infty} (\sqrt[n]{n})^{\pi(n)}.$

The last one seems interesting, since $ \lim_{n \to \infty} (\sqrt[n]{n})=1$, proven here.

How to prove these?

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    Definitions? Of what are those two formulas "definitions"? Anyway, the product of the primes up to n is asymptotic to $e^n$ - I believe this is at the same level of difficulty as the Prime Number Theorem - that should get you the first equation.2012-03-26

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