Karim Belabas on Thu, 20 Nov 2008 04:11:28 +0100

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 Re: Pell's equations and beyond

```* 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]
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