Re: questions about a certain elliptic curve and its points

[Denis Simon <denis.simon@unicaen.fr>]

Hi,

We can generalize a little the idea of Bill:

Let u,a be parameters
for n =  u*(u^2+1)*a^2, we can take
P1=[n*u,u^2*(u^2+1)^2*a^3];
P2=[n/u,(u^2+1)^2*a^3];
P3 = [0,0]
These points satisfy P1+P2+P3 = 2*P3 = 0

The solution of Bill corresponds to u=1.

Notice that the parameter a does not produce a new elliptic curve since
E(u,a) is isomorphic to E(u,1) through urst = [a,0,0,0]

Denis SIMON.

----- Mail original -----
> De: "Bill Allombert" <Bill.Allombert@math.u-bordeaux.fr>
> À: "pari-users" <pari-users@pari.math.u-bordeaux.fr>
> Envoyé: Lundi 5 Mai 2025 22:40:14
> Objet: Re: questions about a certain elliptic curve and its points

> On Mon, May 05, 2025 at 01:06:21PM -0700, American Citizen wrote:
>> Hello:
>> 
>> I have a certain elliptic curve (1)
>> 
>> (1)  E(n) = [0, 0, 0, n^2, 0]
>> 
>> let's say that we pick any 3 points on this curve, P1[x1,y1], P2[x2,y2] and
>> P2[x3,y3]
>> 
>> I am looking for these 3 points to satisfy the relationship in (2) below:
>> 
>> (2)  n^2 = x1*x2 + x1*x3 + x2*x3
>> 
>> where n is positive integer, and x1,x2,x3 are rational or integer.
> 
> If n=2*a^2, then P1[n,4*a^3], P2=P1, P3[0,0] is a solution.
> 
> Cheers,
> Bill.