Base Rep Puzzle

[Michele Dondi]

On Wed, 28 Dec 2005, David Wilson wrote:

> For n >= 2, what is the smallest positive integer that has a different number 
> of digits in each of the bases 2 through n?

For n=2 all numbers have a different number of digits in all bases b, c in 
{ 2 ... 2 } == {2} with b != c, (since there are none such b, c!) so the 
smallest positive one is 1.

For n=3 it is 2.

In the general case the number of digits of N in base b is approximately 
(and it is exactly this "approximately" that makes the 
question interesting!) C(b) := log(N)/log(b). Now the smallest difference 
C(b)-C(b') in absolute value is log(N)(1/log(n-1)-1/log(n)) and it becomes 
1 for

1/log(n-1) - 1/log(n) = 1/log(N).

For large n the left hand member is approximately 1/(n log n)^2 and we 
find the condition

n log n = log(N)^1/2,

this is not solvable in terms of elementary functions either, but may give 
a starting point around which to look at and possibly information about 
asymptotic behaviour, although certainly a more refined analysis would be 
required for both...


Michele
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