From: Clive Tooth <clive@pisquaredoversix.force9.co.uk>
Subject: Re: More on Pi and randomness
Date: Sun, 21 May 2000 19:34:21 +0100
Newsgroups: sci.math
Summary: [missing]

Keith Ramsay wrote:

> In article <8g74kv$rf2$1@slb7.atl.mindspring.net>,
> "r.e.s." <rs.1@mindspring.com> writes:
> || Some things are known about the decimal digits of pi
> || in general. For example, for no positive integer n
> || are the digits n thru 100*n all equal to zero.
> |
> |Some such things are known in general, but I think that
> |that last sentence isn't one of them  -- for if it's
> |true, then pi is not normal in base 10.
> 
> No, if he had said "n through n+100" it would be incompatible with
> normality, but most numbers are both normal and have the property
> above. A normal number has blocks of 100 zeros, but not usually
> starting with the first digit; it does also have blocks of at least
> 99,001 zeros, but the first one usually doesn't start at or before
> the 1,000-th place.
> 
> We know that for some C, for every pair of integers p and q,
> 
>   (1)    |pi-p/q|>C/q^37.
> 
> (In fact we can use something much smaller than 37, but I don't
> remember what the most recent result is. If I remember correctly
> they'd gotten it into the teens.) It is sufficient for Clive Tooth's
> claim to be correct that we always have
> 
>   (2)    |pi-p/10^{n-1}|>10^{-100n}.
> 
> I'm afraid I don't know a constant C which is large enough for any of
> these exponents, but I'm sure that Clive Tooth has given himself
> enough margin for error by using 100 that his claim follows easily
> from what is known. If (1) is true for some C which is not
> fantastically small, for example, it implies that (2) holds for all
> q=10^{n-1} beyond the first few. We already know that (2) holds
> for all n<10^10.

I was relying on theorem 1 in Mahler, K. "On the Approximation of pi"
Nederl. Akad. Wetensch. Proc. Ser. A. 56/Indagationes Math. 15, 30-42,
1953, which is:

  If p and q>=2 are positive integers then |pi-p/q|>q^-42

This paper is reprinted on pages 306-318 of "Pi: A Source Book" by
Berggren, Borwein & Borwein.

The paper (but not the exact theorem) is mentioned at
http://mathworld.wolfram.com/Pi.html

In 1974 Mignotte improved the "-42" to "-20", for sufficiently large q.

In 1984 Chudnovsky and Chudnovsky improved the "-20" to "-14.65", for
sufficiently large q.

-- 
 
Clive Tooth
http://www.pisquaredoversix.force9.co.uk/
End of document