# Re: Digits of pi

*From*: Herman Rubin <hrubin@xxxxxxxxxxxxxxxxxxxx>*Date*: Sat, 7 Jan 2012 18:02:49 +0000 (UTC)

On 2012-01-07, Timothy Murphy <gayleard@xxxxxxxx> wrote:

The Last Danish Pastry wrote:

What is the smallest positive integer, n, such that when n is converted

into its usual string of decimal digits, that string does not occur

anywhere in the first billion digits of the decimal expansion of pi?

I think I know the answer, but I am not totally certain.

I was thinking about this as a problem on probability distributions

(about which I know almost nothing),

and it seemed to me that n would be quite small.

Suppose we consider d-digit numbers.

There are about 10^9 d-digit sequences in the given billion digits.

I'm not at all sure of my reasoning,

but the probability that a _given_ d-digit sequence does not occur

is approximately (1-10^{-d})^{10^9}

If we take d = 9 this is about 1/e.

So it is almost certain that _some_ 9-digit sequence does not occur.

It is certain that some 9-digit sequence does not occur,

as there are only 10^9 - 8 9-digit sequences. However,

this sequence may start with 0, and not give a counterexample.

Your argument just makes it plausible that there is one.

Suppose we take a smaller d.

Let p be the probability that a d-digit sequence does not occur.

Then approximately log p = 10^9 log(1-10^{-d}) = -10^{9-d}.

So p is approximately e^{-10^{9-d}}.

If we take d = 7 this gives p = e^{-100}.

So the probability that some 7-digit sequence does not occur

(this is very crude reasoning) is about 10^7 e^{-100}.

Say 10 = e^3; this gives probability e^{-79},

which is very small, ie all 7-digit sequences almost certainly occur.

If we take d = 8 we get p = e^{-10},

so the probability that some d-digit sequence does not occur

is approximately e^{24-10}, ie it is almost certain.

So my guess is that n = 8.

The probability that the above reasoning is correct is rather small!

--

This address is for information only. I do not claim that these views

are those of the Statistics Department or of Purdue University.

Herman Rubin, Department of Statistics, Purdue University

hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558

.

**References**:**Digits of pi***From:*The Last Danish Pastry

**Re: Digits of pi***From:*Timothy Murphy

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