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?