Re: Pell's equations and beyond

["Max Alekseyev" <maxale@gmail.com>]

Dear Karim,

Thank you for the answers.

One of important reasons I like Dario Alpern's java applet - it simply
does "the job" for generic input by taking care of all possible
branchings and degenerate cases. I would very welcome similar
functionality for PARI/GP...

As PARI/GP provides only basic functionality, I wonder if there is
available any third-party wrapper that would take care of all
degenerate cases in the course of solving general quadratic bivariate
Diophantine equation?

Or is it possible to extend PARI/GP functionality this way?

Thanks,
Max

On Wed, Nov 19, 2008 at 7:09 PM, Karim Belabas
<Karim.Belabas@math.u-bordeaux1.fr> wrote:
> * Max Alekseyev [2008-11-20 00:29]:
>> I dream about having the functionality of Dario Alpern's quadratic
>> bivariate Diophantine equation solver:
>> http://www.alpertron.com.ar/QUAD.HTM
>> in PARI/GP. Is anything like that already present there?
>> At the moment, I'm not even sure if there is a simple way to solve
>> Pell's equations in PARI/GP.
>>
>> Could you please clarify what is the best way (and if there exists one
>> without much programming) to solve the following equations in PARI/GP:
>>
>> 1) Pell's equation x^2 - D y^2 = 1, where D is integer ?
>
> This is more or less given by quadunit(D) or (much better when D is large),
> K = bnfinit(x^2 - D); K.fu
>
> Both assume that D is not a square.
>
>> 2) Generalized Pell's equation x^2 - D y^2 = c, where D and c are integer ?
>
> K = bnfinit(x^2 - D); bnfisintnorm(K, c)
>
> Assumes D not a square. Otherwise, it's "simpler" but a bit tedious to
> program (special case c = 0, otherwise for each divisor of c, you get a
> new 2 x 2 linear system).
>
>> 3) Quadratic bivariate Diophantine equation in the general form: ax^2
>> + bxy + cy^2 + dx + ey + f = 0, where a,b,c,d,e,f are integer
>> coefficients ?
>
> You can reduce it to the above cases by a translation, but it's again
> tedious: have to treat separately a number of degenerate cases...
>
> Cheers,
>
>    K.B.
> --
> Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
> Universite Bordeaux 1          Fax: (+33) (0)5 40 00 69 50
> 351, cours de la Liberation    http://www.math.u-bordeaux.fr/~belabas/
> F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
> `
>