On Aug 7, 3:06?pm, mike3 <mike4...@yahoo.com> wrote: > On Aug 6, 8:18 am, arithmonic <djes...@gmail.com> wrote: > > > Come on, continue by insulting the author of such book (and other > > related people), I am sure that they do not agree with most of my > > critics to the root-solving history and I do not want them to do so, > > but they are true un-biased mathematicians. > > > Come on, persistent offender. > > > Show some precedents on the methods shown in my webpages. > > Ignore the guy's childish insults, drop your > OWN childish insults and RESPOND TO HIS POINT: > > "Generalized continued fractions should exhibit > the same properties as simple continued fractions. > SCFs produce best rational approximations, > generalized continued fractions should produce best > simultaneous approximations.
It is not at all clear to me that this is possible. See the paper "Best Simultaneous Diophantine Approximations II. Behavior of Consecutive Best Approximations" by J.C. Lagarias. It seems as though something has to be relaxed in the generalization.
> If the rational mean can produce generalized > continued fractions, it should produce all > best simultaneous rational approximations. > If it can do this it can then solve the cubic Pell, > whose solutions are best rational simultaneous > approximations to cubrt(k*k), cubrt(k). " >
Finding solutions to the cubic Pell is easy (use Pari/gp). Proving (unconditionally) the solution you find is, say, the fundamental solution can take a bit more effort.