I think that I have realized why you insist so much in talking about best approximations and pell's equation. Another webpage from mine show a brief description on new "Generalized continued Fractions":
the esence of such new GCF is to show that one can easily construct periodic representations of irrational numbers of higher degree tha 2, i.e., periodic representations of non-quadratic roots.
In this way, if one try to represent the cube root of 2 by means of the traditional continued fractions (Second order continued fractions as we should call them) then one get a distorted representation (non- periodic coefficients) of this irrational number. By means of these new GCF one can find periodic representation of irrational numbers of degrees higher than 2.
Some time ago I posted some info on that webpage and it might be you understood that I was claiming to have solved the Cube version of Pell's equation. Of course, I must say that the new ARITHMONC MEAN (shown in my webpages) could be a very useful tool to study the issue on Cube version of Pell's equation.
But your statement about that I claimed to have solved the cube Pell's equation is another absolute FALSE STATEMENT from yours. Of couse I always grant people a second chance, it might be that you misunderstood all what I told about GFC or some other people intentionally caused confusion as it has been the habit for some people who try to prevent others from looking at the new methods shown in my webpages becasue of my critics to the whole history of root-solving, I really don't care, these simple methods will find their way all through young minds, and no one will be able to stop that. I do not look to get favors from any peer-reviewed journal in exchange for not including my critics against the history of root-solving. My criticism on the history of root-solving will continue for so long time.
Anyway, notice that I will not discuss GCF this thread, just take this as a marginal note.