Re: Digits of pi
 From: Herman Rubin <hrubin@xxxxxxxxxxxxxxxxxxxx>
 Date: Sat, 7 Jan 2012 18:02:49 +0000 (UTC)
On 20120107, 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 ddigit numbers.
There are about 10^9 ddigit sequences in the given billion digits.
I'm not at all sure of my reasoning,
but the probability that a _given_ ddigit sequence does not occur
is approximately (110^{d})^{10^9}
If we take d = 9 this is about 1/e.
So it is almost certain that _some_ 9digit sequence does not occur.
It is certain that some 9digit sequence does not occur,
as there are only 10^9  8 9digit 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 ddigit sequence does not occur.
Then approximately log p = 10^9 log(110^{d}) = 10^{9d}.
So p is approximately e^{10^{9d}}.
If we take d = 7 this gives p = e^{100}.
So the probability that some 7digit 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 7digit sequences almost certainly occur.
If we take d = 8 we get p = e^{10},
so the probability that some ddigit sequence does not occur
is approximately e^{2410}, 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)4946054 FAX: (765)4940558
.
 References:
 Digits of pi
 From: The Last Danish Pastry
 Re: Digits of pi
 From: Timothy Murphy
 Digits of pi
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