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

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

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David Bernier

Posts: 3,242
Registered: 12/13/04
might one improve on Hurwitz' Theorem for Diophantine approximations
to pi?

Posted: Apr 5, 2013 4:17 AM
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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) ...


David Bernier

--
Jesus is an Anarchist. -- J.R.



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