Digit-Shifting Constants
Given a real number 
, find the powers
of a base 
that will shift the digits of 
a number of places
to the left. This is equivalent to solving
 |
(1)
|
or
 |
(2)
|
The solution is given by
 |
(3)
|
where 
is the Lambert
W-function.
The above plot shows 
for 
and small values of 
. As can be seen,
there are two distinct solutions, corresponding to two different branches
of 
in (3). For 
, 2, ..., these solutions are approximately
given by 0.137129, 0.0102386, 0.00100231, 0.000100023, 0.0000100002, ..., and 1,
2.37581, 3.55026, 4.66925, 5.76046, ..., respectively. For example,
 |
(4)
|
and
 |
(5)
|
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Apéry's constant approximations