On Wed, 14 Aug 2013, 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?
I don't know how rare it is, but it's certainly not impossible. For example, the continued fraction for .7501324981324 is [0 1 3 471 2 5 37 1 217 1 6 5 1 2 4 5 5] (or my program has a bug.) The first 3 terms are [0 1 3]=3/4 (and the next term is huge, so this is a pretty good approximation.) Of course sqrt(3^2+4^2)=5. I arrived at .7501324981324 by typing .750 and then banging on my keyboard. I think that every Pythagorean fraction has an interval containing it where the fraction itself is a convergent of the continued fractions of all the numbers in the interval. Without thinking about how to prove it, I bet that the density of those intervals is non-zero and that Pythagorean approximants aren't infinitely rare. But I've been wrong before... -- Tom Duff. I was typing and suddenly my laptop was in Iran.