Re: questions about a certain elliptic curve and its points

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

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.