Re: Arbitrary strings of digits in the decimal display of Pi

From: Carl Devore (devore_at_math.udel.edu)
Date: 09/20/04 

Date: Sun, 19 Sep 2004 21:08:39 -0400

On Mon, 20 Sep 2004, J.Barsuhn wrote:
> It appears to be generally accepted that one can find any sequence of
> digits somewhere in the decimal display of Pi = 3.14159....

It is generally accepted because the probabilty is 1 that a randomly
selected real number will have this property.

> Does anybody know whether a proof of this conjectured property has been
> given (or attempted) ?
> Are there other irrational numbers that are expected to exhibit this
> same property?

That is a major unsolved problem. See
http://mathworld.wolfram.com/NormalNumber.html. A Google search on
"normal numbers" and "absolutely normal numbers" will provide much
information also.

> Of course, this cannot be a general property of irrational numbers.

No, but it must be of most of them. Consider these related problems:

1. Let F(n) denote the number of positive integers less than n that do not
contain digit 5 in their decimal expansion. What is the asymptotic
expansion F(n)/n as n -> infinity? Does this change substantially if 5 is
replaced by a longer string of digits?

2. Does the sum of the reciprocals of the positive integers that do not
contain digit 5 in their decimal expansions converge?

3. What is the sum of the reciprocals of all p-smooth positive integers?

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