You should look at semi convergents. Look at http://shreevatsa.wordpress.com/2011/01/10/not-all-best-rational-approximati... for a good account. Victor Sent from my iPhone On Aug 14, 2013, at 10:23, Henry Baker <hbaker1@pipeline.com> wrote:
Suppose I'm trying to compute some rational approximation to a real number x, but with some additional constraint.
In particular, if m/n approximates x, I'd like sqrt(m^2+n^2) to be integral.
Suppose I used a continued fraction process to generate better & better approximations.
Is there any reason to believe that I'd eventually find one m/n for which sqrt(m^2+n^2) is integral ?
I did a quick search on both pi and e, and so far _none_ of the rational approximations (except for early integral approximations) m/n has sqrt(m^2+n^2) integral.
So this leads me to believe that perhaps what I'm trying to do is impossible; perhaps sqrt(m^2+n^2)=integer _never_ happens for rational approximations produced by continued fractions?
Or perhaps this situation is exceedingly rare.
If so, how rare is it?
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun