Re: Zero digits in powers

*** T. Winter wrote:
> In article <15364-42B76153-906@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> ol3@xxxxxxxxx (Oscar Lanzi III) writes:
> > *** T. Winter makes a valid point. When you use a probabilistic model,
> > you're really guessing when you have accurately identified a sample
> > space. In the fifth-power summation example, the factorization in the
> > formula happesn to give enough information for the proabilistic model to
> > identify the right proposition.
>
> Yes, in that case. However, there is *no* way that it shows that it
> actually *is* the right proposition.
>
> > Probabilistc models can give an idea of
> > what might be most fruitful to try to prove -- but if in the process of
> > solving the problem you come across new information, you need to
> > reevaulate your probabilistic expectations in light thereof.
>
> Indeed. The probabilistic method only gives an indication of what might
> be true. It can not replace a proof.
>

Sadly this is definitely the case.

>
> > Incidentally, 34N^2-1 is always congruent with 5 or 7 modulo 8. There
> > you have it.
>
> Indeed. Nothing probabilistic about it.
>

And N^2 is always congruent with 0 or 1 modulo 3. So what? I stopped
geting surprized by such facts back in the 6th grade.

How does that restrict the occurrence of a 0 somewhere in the middle of
the decimal representation of a long power of 2?

As I said, middle digits in products of long numbers have been used by
quite competent mathematicians to generate pseudo-random numbers.

>
> And back to the original. When you are looking at powers of 2, decimally
> written, I do not think there is a probabilistic proof strong enough to
> show that there is a largest power where there is no 0 in the expansion.
>

That's both true and obvious.

>
> 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?

.

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