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Re: Looking for who originally conjectured the following theorem
Posted:
May 23, 2009 3:49 PM
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Rainer Rosenthal wrote:
> alainverghote@gmail.com schrieb:
> > On 23 mai, 10:37, Rainer Rosenthal <r.rosent...@web.de> wrote:
> >> alainvergh...@gmail.com schrieb:
> >>
> >>> but I found interesting to build a method for cases,
> >>> Example: n = 71 , 2^p , p?
> >> p=149, 2^p=713623846352979940529142984724747568191373312.
> >>
> >> Here is another easy one: n = 2, 71^p, p?
> >>
> >> Solution: p=4, 71^p=25411681.
> >>
> >> Cheers,
> >> RR
> >
> > Bonjour Rainer,
> >
> > Well, why writing "easy one" if you do not mind
> > explaining your way ?
>I am going to post this under the natural number thread. My methods will be explained and out of the Einstein box far more interestring than standard stock Maple. Cheers, MMM(¤¤¤) I'll hit you with a link. Thank you.
> Sorry, but it's not an especially interesting method.
> Just checking ranges [n*10^m .. (n+1)*10^m-1] whether or
> not there is some b^k within this range, b being the base
> in question.
> The "easy" adjective is referring to the low exponent k,
> resulting in a very short run of the search program (Maple):
>
> WhichPowerB := proc(n,b) # k: leadDig(b^k) = allDig(n)
> local m,erg,k;
> erg := 0;
> for m from 0 to infinity do
> k := floor(evalf(log[b](n*10^m)));
> if n*10^m <= b^k and
> b^k <= (n+1)*10^m-1 then
> erg := k;
> break;
> elif n*10^m <= b^(k+1) and
> b^(k+1) < (n+1)*10^m-1 then
> erg := k+1;
> break;
> fi;
> od:
> print(n,erg,b^erg);
> return;
> end:
>
> For n = 1, 2, ... up to 10 we get:
>
> for n to 10 do
> WhichPowerB(n,2);
> od;
>
> n k 2^k
> -------------------
> 1 0 1
> 2 1 2
> 3 5 32
> 4 2 4
> 5 9 512
> 6 6 64
> 7 46 70368744177664
> 8 13 8192
> 9 53 9007199254740992
> 10 10 1024
>
> For your question regarding n=71:
>
> WhichPowerB(71,2);
> 71, 149, 713623846352979940529142984724747568191373312
>
> The "easy" one with base 71:
>
> WhichPowerB(2,71);
> 2, 4, 25411681
>
> (For n=2 and base b=10 one gets into trouble :-) )
>
> Best regards,
> Rainer Rosenthal
> r.rosenthal@web.de
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