From: israel@math.ubc.ca (Robert Israel) Newsgroups: sci.math Subject: Re: conjecture on weird series Date: 1 Mar 1996 01:49:14 GMT In article <4h30oc\$sv1@nntp.ucs.ubc.ca>, israel@math.ubc.ca (Robert Israel) writes: |> For every irrational number x there are infinitely many m/n with |> |x - m/n| < 1/n^2. I'm not sure about the restriction to n odd, but I |> believe that can be handled as well. Yes, W.T. Scott (Bull. Amer. Math. Soc. 46 (1940), 124 - 129) showed that for every irrational number x there are infinitely many m/n with |x - m/n| < 1/n^2 and m/n in any one of the categories odd/odd, even/odd and odd/even. -- Robert Israel israel@math.ubc.ca Department of Mathematics (604) 822-3629 University of British Columbia fax 822-6074 Vancouver, BC, Canada V6T 1Y4 ============================================================================== From: kubo@abel.harvard.edu (Tal Kubo) Newsgroups: sci.math Subject: Rational Approximations to Pi (was Re: conjecture on weird series) Date: 2 Mar 1996 08:40:12 GMT Robert Israel wrote: > > [re: solutions m,n of |Pi - m/n| < c/n^q, for c>0 and n>>0] > > The best bound that has been proven, as far as I know, is that > with q=42 and some c there are no solutions (K. Mahler, Indag. > Math. 15 1953) 30-42). This comes up often enough that it may warrant FAQ'ing. The record irrationality measures to date (ie, smallest known replacements for '42' above) are: for Pi: 8.016045... M. Hata, Acta Arithmetica 63 (1993) 335-349 Pi^2: 7.81 | D.V Chudnovsky & G.V Chudnovsky, | Approximations and Complex Multiplication Pi*sqrt(2): 16.67 | According to Ramanujan, in _Ramanujan | Revisited_ (Urbana-Champaign 1987), 375-472 Pi*sqrt(3): 5.8 | publ. Academic Press, 1988