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

[Max Alekseyev <maxale@gmail.com>]

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 <lael.cellier@laposte.net> 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.