# [seqfan] Re: An arithmetic conjecture

Fri Mar 6 21:27:30 CET 2009

```Hallo all!

DW> To my knowledge, no one knows how to prove your statement.
DW> Without getting heavily into it, this problem belongs to a class
of problems
DW> with problems like:

DW> Does every sufficiently large power of 2 include the digit 0 in base 10?

DW> which statistically are true with probability 1, but have not to my
DW> knowledge been proved.

Now I see that Wolfgang Kirschenhofer in the newsgroup discussion
did exactly that: He reduced the conjecture to:

"If every sufficiently large power of 2 include the digit 0 in base 3 ..."
plus "The number of digit 1 at lower positions is odd."

This was a key observation when we looked for a fast algorithm
to check the conjecture numerically.

DW> I could go more deeply into this problem. I have a general conjecture
DW> implying that all sufficiently large m satisfy [2^m / 3^k] mod 6 = 3 for
DW> some k > 0, but I cannot prove this conjecture nor can I vouch for
m = 26 as
DW> the largest exception.

DW> Explaining my conjecture is too involved for a late night email. I might go
DW> into it if there is sufficient interest from seqfan (or Tanya).

I can assure from the discussion in the newsgroup that this
problem attracts some interest. I would be very thankful if
you would explicate your general conjecture and/or give
pointers to the literature.

The sequence is submitted and will (hopefully) appear as A157409.

Cheers Peter

```