Re: Zero digits in powers




On 21/06/2005 19:55, vkarlamov@xxxxxxxxx wrote:

> *** T. Winter wrote:
>> In article <1119321075.559504.308790@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
>> vkarlamov@xxxxxxxxx writes:
>>> *** T. Winter wrote:
>> ...
>>>> These kinds of (number theoretical) problems are beyond the capacities of
>>>> probability.
>>>
>>> But are these problems NOT beyond the capacities of number theory? Or
>>> do you want to have another FLT on your hands? How long will it take
>>> before a wiles finally proves it in year 2350?
>>
>> That does not matter. There are enough open problems in number theory.
>> Some of those may be solved at some time. But possibly some of them
>> may never be solved. Also FLT itself was not very important or even
>> interesting. It was the mathematics developed to solve it that is
>> really interesting.
>>
>
> Tell me, ***, why do you guys think that your hypothesis is true? What
> evidence do you have to support it?
>
> I am not being festidious here. I am in the process of making a n
> important point. So please answer.
>
> Describe to me the whole scope of your (empirical?) evidence. For which
> values of N did you discover that it holds?

Are you talking about the OP ?

1. Have the idea to look for zero (maybe too drunk to think about anything
else ;-))

2. Try for some values by hand. To clarify things a bit, try more values
(say, 1000), with a computer.

3. Discover some possible limits: 2^86 and n^40 here

4. Mmmm. Many digits, no evidence that they make a pattern. So statistical
*clue* that it might be right

5. Try many more values. Say up to 10^9

6. Well... quite interesting after all, maybe it's worth asking if somebody
have studied that before.

Please notice one important thing: statistical clue is by no way a proof,
it's just a feeling that things might be simple. There are problems where it
is proven that there are solutions, but it is also proven that the smallest
is larger than what a computer can handle. So statistical clue, even for
tests up to 10^20 if you can do that, is of no value at all.

For the moment, the only idea I had was looking for repeating patterns.
It worked for n^2500 with low order digits, maybe there are clever pattern
in high order digits or in the middle. Maybe there is another approach, but
please, if you want to help, I'm only interested in a *proof*. I think
probabilities could help only for one question here: find a law that digits
follow approximately, and derive a probability for finding a counter example
in [M,N], for given M and N, assuming there are indeed counter examples.
But I doubt one could derive small M and N too look at from this study.

Actually, I believe the result is true for statistical reasons, and I know
perfectly well these are very bad reasons in number theory. I would not be
surprised to find counter examples.



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