[seqfan] Numbers of the Form 2^j (mod 10^k)

[Edwin Clark]

On Fri, 27 Aug 2004, Paul D. Hanna wrote:

>      Given only the last 5 (say) digits of a large integer N, 
> how can you determine if N is NOT some power of 2? 
> That is, which 5-digit numbers are of the form: 
>     2^j (mod 10^6) 
> where j is any positive integer? 

Paul,

You may be interested in a sort of related discussion that took place 
not long ago on sci.math:

Subject: Re: Power of 2 with all even digits?
Newsgroups: sci.math
Date: 2004-07-03 17:45:04 PST 

This thread discussed the distribution of the digits of 2^n for large n. 
As I recall it seems that for large n the digits appear to be pretty  
randomly distributed. 

For example, in 2^n where n is 1 million, Maple gives the following 
character frequences:

  "0" = 30186, "1" = 30354, "2" = 30047, "3" = 30193, "4" = 30230,

  "5" = 30174, "6" = 30103, "7" = 29840, "8" = 29896, "9" = 30007

The same near uniformity holds for successive pairs of digits in 2 to the 
1 million. 

Which suggests that it is unlikely that a simple description 
of the k-digit numbers of the form 2^j mod 10^k can be found.

Edwin