Relationship between $(2^n)^{th}$ prime and its bits

Asked Jul 10 '21 at 13:25

Active Jul 10 '21 at 13:25

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How many bits does it take to write the $p_{2^n}$, the $(2^n)^{th}$ prime number in binary where the first one is 2, second 3, third 5 et cetera.

Is it about $n$? E.g., if I take the product $p_{2^{25}}p_{2^{50}} = x$ then will $x$ be approximately $25+50=75$ bits?

asked Jul 10 '21 at 13:25

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Hint : Use Pierre Dusart's bounds for the $n$ th prime number. – Peter Jul 10 '21 at 14:00
This is not a homework problem, I could set-up my experiment better if I can tell this efficiently. Could I get more than a hint? – scribe Jul 10 '21 at 14:04
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$n\cdot 2^n\cdot \log 2$ has how many bits? – Roddy MacPhee Jul 10 '21 at 21:50
Assuming $\log = \log_2$, we have about $\log_2(n) + n + 1$ bits? – scribe Jul 12 '21 at 11:02

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