Max Alekseyev on Tue, 25 Nov 2008 22:16:06 +0100

 Re: Pell's equations and beyond

```On Fri, Nov 21, 2008 at 6:45 AM, Bill Allombert
<Bill.Allombert@math.u-bordeaux1.fr> wrote:

>> 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
>
> You are slightly unfair with PARI: PARI includes efficient algorithms to solve
> this task in (at worse) subexponential time, and so it can be used to deal with
> much larger coefficients than the above applet, and that is the hard
> part of the work.

I did not mean to diminish PARI/GP abilities. I just wanted to check
if there is an extension that will make its functionality more "user
friendly" w.r.t. solving quadratic diophantine equations.
The aforementioned applet is user friendly but far from optimal and
has certain limitations (e.g., on the size of coefficients/solutions).
That's why I wondered if PARI/GP can provide similar "interface" to
the user.

On the other hand, I don't feel myself experienced enough neither with
PARI functionality nor with the theory of quadratic forms to write an
efficient GP script. That's was the reason to request it here.
Anyway, thank you for the script, even though it does not handle all the cases.

P.S. btw, I have a somewhat related question - what is the most
efficient way to solve in PARI/GP equations of the form a*x^2 + b*y^2
+ c*z^2 = 0 w.r.t. integer x,y,z, where a,b,c are given integer
coefficients?

Regards,
Max

```