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Old 2013-11-29, 15:40   #1
jnml
 
Feb 2012
Prague, Czech Republ

132 Posts
Default [Curiosity] Binary logarithm of a Mersenne number

The binary logarithm[0] L of a Mersenne number M_n,\ n \in N, having enough precision to
reconstruct M_n exactly after rounding 2^L to an integer, ie. \left| M_n - 2^L \right| < {1} \over {2} is

\ \ \ \ \ L = n - 2^{1-n}.

----

The integral part of L is n-1. The fractional part of L consists of n-1 binary ones.

For example:

Code:
n	L            L (base 2)
-------------------------------
1       0            0
2	1.5          1.1
3	2.75        10.11
4	3.875       11.111
5	4.9375     100.1111
...
[0]: http://en.wikipedia.org/wiki/Binary_logarithm
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