mersenneforum.org - View Single Post - [Curiosity] Binary logarithm of a Mersenne number

jnml · 2013-11-29, 15:40

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

2013-11-29, 15:40	#1
jnml Feb 2012 Prague, Czech Republ 13² Posts	[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