Re: Is it possible to have several solutions in this way to this equation using Pari/ɢᴘ ?

You define alpha and beta as
beta=-(V\W);
alpha=W*(V+W*beta);
and then you say that you want to find them. Please clarify.

As for xx having two integer zeros, it can stated as equation
( alpha^2*x^2+(2*alpha*beta-f*b)*x+(beta^2-c) ) - (
alpha^2*y^2+(2*alpha*beta-f*b)*y+(beta^2-c) ) = 0
to be solved in distinct integers x,y. Cancelling nonzero factor x-y, we get
alpha^2 * (x+y) = -(2*alpha*beta-f*b).
That is, existence of two integer zeros translates into
(2*alpha*beta-f*b)/alpha^2 being an integer.

Regards,
Max

On Sun, Jan 19, 2025 at 10:28 AM Laël Cellier <[email protected]>
wrote:

> Not exactly as there’s no second equation mais un polynome…
>
> Currently the script for solving this is :
>
> beta=-(V\W);
> alpha=W*(V+W*beta);
> xx=alpha^2*x^2+(2*alpha*beta-f*b)*x+(beta^2-c);
> nfr=nfroots(,xx);
>
> So given v,w,b,c I want to find integers alpha and beta such I can find at
> least 2 different but valid values of nfr. I think this means modifying the
> part of script for finding alpha and beta. This might not even be
> mathematically possible at all…
> Cordialement,
>
> Le 19/01/2025 à 15:07, Bill Allombert a écrit :
> > On Sun, Jan 19, 2025 at 11:58:47AM +0100, Laël Cellier wrote:
> >> Bonjour,
> >>
> >> I’ve the following equation where the aim is to find /alpha/ and /beta/
> as
> >> integers given /w/ and /v/ as integers
> >>
> >> alpha == w (v + w beta)
> >> Of course finding several solution for the equation above is possible,
> but
> >> then I want /nfroots()/ to return a second set of possible results
> given /c/
> >> and /b/ and where /x/ is an unknow
> >>
> >> xx=alpha^2*x^2+(2*alpha*beta-abs(b))*x+(beta^2-c);
> >> nfroots(,xx);
> > So given v,w,b,c you want to find integers alpha, beta and rational x
> such that
> >
> > alpha = w *(v + w * beta)
> > alpha^2*x^2+(2*alpha*beta-abs(b))*x+(beta^2-c) = 0
> >
> > Is it correct ?
> >
> > Cheers,
> > Bill.
>
>