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Topic: Looking for who originally conjectured the following theorem
Replies: 24   Last Post: Jun 9, 2009 9:45 PM

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 Rainer Rosenthal Posts: 963 Registered: 12/6/04
Re: Looking for who originally conjectured the following theorem
Posted: May 23, 2009 1:19 PM

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

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

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

Date Subject Author
5/21/09 hierholzer@reborn.com
5/21/09 Gerry Myerson
5/22/09 alainverghote@gmail.com
5/22/09 gerry@math.mq.edu.au
5/23/09 alainverghote@gmail.com
5/23/09 Rainer Rosenthal
5/23/09 alainverghote@gmail.com
5/23/09 Guest
5/23/09 Rainer Rosenthal
5/23/09 Axel Vogt
5/23/09 Denis Feldmann
5/23/09 Rainer Rosenthal
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5/27/09 alainverghote@gmail.com
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