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Topic: cube root of a given number
Replies: 112   Last Post: Jan 10, 2013 1:39 PM

 Messages: [ Previous | Next ]
 Iain Davidson Posts: 1,173 Registered: 12/12/04
Re: cube root of a given number
Posted: Aug 11, 2007 2:53 PM

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.

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 ?

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