Date: Apr 7, 2013 8:28 AM
Author: David Petry
Subject: Re: might one improve on Hurwitz' Theorem for Diophantine<br> approximations to pi?
On Friday, April 5, 2013 1:17:29 AM UTC-7, David Bernier wrote:
> Hurwitz' Theorem on Diophantine approximations states that,
> if alpha is an irrational number in the reals R,
> then for infinitely many positive integers m,n with
> gcd(m, n) = 1, one has: |alpha - m/n| < 1/(sqrt(5)*n^2) .
> < http://en.wikipedia.org/wiki/Hurwitz%27s_theorem_%28number_theory%29 > .
> Do number theorists contemplate as "somewhat possible"
> that for alpha=pi, one might be able to prove a bit
> more without a 10+ year effort by many, i.e.
> an improvement by epsilon without huge effort?
> The improvement would go like this:
> |pi - m/n| < C/n^2 for infinitely many coprime positive
> integers m, n for a stated C (e.g. "C = 1/sqrt(5) - 1/10^100." ),
> with C < 1/sqrt(5) ...
I believe it's well known that the constant C = 1/sqrt(5) is only required for special numbers closely related to the golden ratio. For all other numbers, a considerably smaller constant applies.