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

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 r3769@aol.com Posts: 352 Registered: 12/12/04
Re: cube root of a given number
Posted: Aug 11, 2007 5:10 PM

On Aug 11, 11:53?am, "sttscitr...@tesco.net" <sttscitr...@tesco.net>
wrote:
>
> 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 ?
>

What I had in mind was something like this:

(12:39) gp > ?bnfinit
bnfinit(P,{flag=0},{tech=[]}): compute the necessary data for future
use in
ideal and unit group computations, including fundamental units if they
are not
too large. flag and tech are both optional. flag can be any of 0:
default, 1:
insist on having fundamental units, 2: do not compute units, 3: small
bnfinit,
which can be converted to a big one using bnfmake. See manual for
details

(12:39) gp > a=bnfinit(X^3-25,1);

(12:39) gp > a.fu
%34 = [Mod(4/5*X^2 - 2*X - 1, X^3 - 25)]

(12:41) gp > p3(k,x,y,z)=x^3+k*y^3+k^2*z^3-3*k*x*y*z

(12:41) gp > p3(25,-1,-2,4/5)
%35 = -1

I also have a homebrewed method for finding solutions that is pretty
fast (it finds a 30000+ digit solution for k=1000700 in less than a
minute) but comes without any guarantees about the solution being
fundamental. Pari is much faster, more general, and comes with a
conditional guarantee.

> 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 ?

No, not really. Moveover, given any reasonable definition of "a small
number of trials" I believe it is possible to show the approach I had
in mind does not work. I have not played with it much, however.

Here is a link to an article on this subject you may find interesting:

Determining the Fundamental Unit of a Pure Cubic Field Given Any Unit
N. S. Jeans; M. D. Hendy
Mathematics of Computation, Vol. 32, No. 143. (Jul., 1978), pp.
925-935.
Stable URL:

Rich

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