# [seqfan] Re: An arithmetic conjecture

Jim Nastos nastos at gmail.com
Mon Mar 16 18:48:31 CET 2009

```On Mon, Mar 16, 2009 at 9:20 AM, Rainer Rosenthal <r.rosenthal at web.de> wrote:
> David Wilson wrote:
>> 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?
>>
>
> I'd like to ask my preferred book UPINT, but I am not sure
> where I could find an answer (or at least related questions).

The Google Books copy of UPINT has the section "F24: Some Decimal
Digit Problems:"
where it says: "The decimal rep of 2^n contains no zero digit for
n=1,2,3,4,5,6,7,8,9,13,14,15,16,18,19, ... 86.
Shall we ever know if this sequence is complete? Dan Hoey has checked
that there are no others with n <= 2.5billion "

JN

> The actual problem, posed by Peter Luschny, was: what can
> be said about the remainders of floor(2^m/3^k) modulo 6?
> His problem restated:
>
>    ================= Conjecture ===================
>
>    For all m > 26 there exists some k > 0 such that
>             floor(2^m / 3^k) = 3 (mod 6).
>
>    ================================================
>
> Could someone please be so kind as to give me a pointer
> to UPINT (Unsolved Problems in Number Theory)? I have
> the third edition available at home.
>
> Best regards,
> Rainer
>
>
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>
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>

```

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