On 11 Aug, 17:47, rich burge <r3...@aol.com> wrote: > On Aug 7, 3:06?pm, mike3 <mike4...@yahoo.com> wrote: > > > On Aug 6, 8:18 am, arithmonic <djes...@gmail.com> wrote: > > "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.
Yes, that's the point. The OP was claiming that he had invented a form of generalized continued fraction the produced best simultaneous ? rational approximations and was periodic for higher irrationalities. It turns out that he was not claiming that he could produce all best rational approximation but simply that the occasional best approximation might occur among the "convergents".
He now says that his GCFs produce periodic expansions of higher irrationalities in some way based on Bernoulli's dominant zero method.
But this is not what is usually meant by periodicity. If every algebraic number has a "periodic expansion" in the sense that it is some function of the coefficients of its characteristic polynomial, then there is no way of discriminating between irrationalities of various degrees.
Essentially, all the OP is doing is expressing the ratio of the nth and n-1th terms in a recurrence relation as a complicated fraction involving the coefficients of the characteristic polynomial. I would not have thought that this approach would produce periodicity, except in this trivial sense, or any best rational approximations.
Finding solutions to the cubic Pell is easy (use Pari/gp).
Do you mean systematically using generalizations of continued fractions or other methods for finding cubic units or just intelligent trial and error ?
Proving (unconditionally) the solution you find is, say, the fundamental solution can take a bit more effort.
Yes, I saw your recent post and Israel's interesting answer. Have you found some way of estimating an upper bound for the power of the unimodular matrix so that you can find the fundamental unit after a small number of trials ?