
Re: cube root of a given number
Posted:
Aug 11, 2007 7:11 PM


On 11 Aug, 22:10, rich burge <r3...@aol.com> wrote: > 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'm not familiar with Pari/gp, but there are methods using the regulator and cycles of ideals that always find fundametal units. Aren't they decribed in Cohen's book ? What's the basic idea ? Isn't this a long way from (vector) continued fraction algorithms ?
> 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.
Yes, you have mentioned this before, but were not specific. Didn't you say it was a probabilistic version of Jacobi erron ? > > 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.
Thanks, that looks very interesting.

