
might one improve on Hurwitz' Theorem for Diophantine approximations to pi?
Posted:
Apr 5, 2013 4:17 AM


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.

