This is a very interesting question. We know that the rational points on the unit circle are all of the form (a/c, b/c) where (a,b,c) is a Pythagorean triple. And we know these are dense in the unit circle, so this approximation must be possible. (In fact these rational points form an interesting subgroup of the unit circle group SO(2).) --Dan On 2013-08-14, at 10:23 AM, Henry Baker 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?
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