Thanks very much! This excellent draft paper by Cortzen is referenced in the discussion mentioned by Victor: From: http://lanco.host22.com/web_documents/continfrac.ver.4.pdf Size: 247 KB (252,129 bytes) At 10:40 AM 8/14/2013, Victor S. Miller wrote:
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?