Denis Simon on Thu, 27 Nov 2008 11:21:08 +0100 |
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Re: Pell's equations and beyond |
Dear Max, In order to find a rational solution for the equation a*x^2+b*y^2+c*z^2=0, or more general quadratic equations, you can use my GP script: - step 1: download it from www.math.unicaen.fr/~simon/qfsolve.gp - step 2: start GP - step 3: type the command \r qfsolve.gp (eventually with the full path to this file) - step 4: build the Gram matrix Q of the quadratic form. In your situation just type : Q = matdiagonal([a,b,c]); - step 5: Solve the equation : sol = Qfsolve(Q) -step 6: the answer is either a rational solution or a prime number such that the equation has no local solution at that prime. Now, have fun with it ! Denis SIMON. On Tue, 25 Nov 2008, Max Alekseyev wrote: > 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 >