Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math

Topic: might one improve on Hurwitz' Theorem for Diophantine approximations
to pi?

Replies: 3   Last Post: Apr 7, 2013 1:38 PM

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
David Petry

Posts: 1,103
Registered: 12/8/04
Re: might one improve on Hurwitz' Theorem for Diophantine
approximations to pi?

Posted: Apr 7, 2013 8:28 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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) .


> Cf.:
> < 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.



Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.