
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 about tech.
(12:39) gp > a=bnfinit(X^325,1);
(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^33*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. > > 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 ?
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. 925935. Stable URL: http://links.jstor.org/sici?sici=00255718%28197807%2932%3A143%3C925%3ADTFUOA%3E2.0.CO%3B20
Rich

