Re: Zero digits in powers

Jean-Claude Arbaut wrote:
>
> Le 17/06/2005 03:55, dans
> 1118973329.728644.184320@xxxxxxxxxxxxxxxxxxxxxxxxxxxx,
> « vkarlamov@xxxxxxxxx » <vkarlamov@xxxxxxxxx> a écrit :
>
> > Jean-Claude Arbaut wrote:
> >> Le 17/06/2005 03:35, dans
> >> 1118972154.204248.69090@xxxxxxxxxxxxxxxxxxxxxxxxxxxx,
> >> « vkarlamov@xxxxxxxxx » <vkarlamov@xxxxxxxxx> a écrit :
> >>
> >>> I vaguely recall a story that in the 1960s some Russian programmers
> >>> were using middle digits from products of two big numbers as a "random
> >>> number generator".
> >>
> >> Maybe you'll want to have a look at this thread "Manansala Random Number
> >> generator". I believe there was a discussion on that topic, at one point.
> >>
> >
> > Heck, I am just trying to give productive suggestions as to how to
> > explain the phenomenon in this thread: zeros in 2^n.
> >
>
> Sorry :-)
>
> It's perfectly understandable that numbers with as many as 10000 digits
> with no particular a priori digit distribution property, will have some
> "0" digits in them. But what is *very* interesting is a proof ;-)
> Anybody playing some time with the problem will believe the property
> is reasonnably true, that's another matter to find a rigorous justification.
>

But the approach that I am suggesting not only provides the intuitive
reason why there should be at least one 0 in high powers of a given
number, but gives some promising leads for giving a rigorous proof.

For example, if one could find a lower bound on the probability of 0
happening among "inner" digits and showed that this bound isn't
"conditional" on doccurrence of 0 in other positions, one could find a
rigorous proof.

More than that, I truly believe that the fact that there should be a 0
inside the decimal representation of very long numbers isn't a question
in number theory. It is truly a question in probability and only the
probailistic aaproach can give a general proof.

.

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