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Given the number $n$. Why does it take maximum of $\frac{n}{\log n}$ digits to represent the number $2^n - 1$ in base of $n$?

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Why is this listed as complex analysis? – Thomas Andrews Dec 13 '12 at 18:51
Thanks Thomas Andrews! – user1901968 Dec 13 '12 at 18:51
Because it is a part of a complexity algorithm called Radix-Sort – user1901968 Dec 13 '12 at 18:52
Complexity is not the same as complex analysis, which is analysis on the complex plane. – Thomas Andrews Dec 13 '12 at 18:53
Changed the tag :) – user1901968 Dec 13 '12 at 18:54
up vote 0 down vote accepted

Hint: to express any number $m$ in base $n$ takes $\lfloor \log_n m \rfloor +1$ digits. To see this, think about base $10$. With one digit you can express any number up to $9$, for which the floor of the log is $0$. With three digits you can express numbers up to $999$, for which the floor is $2$. What base of logs were you planning to use?

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Helped a lot! Thanks Ross! – user1901968 Dec 13 '12 at 19:07

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