From: Clive Tooth 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." 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