Re: interesting discovery about elliptic curve [0,0,0, 393129,0]

[American Citizen <website.reader3@gmail.com>]

This might not be so unusual after all. I set up other [0,0,0,n^2,0] 
curves and found that most had the x-coordinates a rational square.

Curious as to why?

Randall

On 10/23/24 17:02, American Citizen wrote:
> To all:
>
> While working with Heron triangles with 2 square sides (the 3rd is not 
> necessarily a square) I came across an interesting elliptic curve
>
> E = [0, 0, 0, 393129, 0]
>
> I checked the first 618 rational points on this curve (sorting by 
> height) and every x-coordinate is a square.
>
> Can anyone explain why this rank 3 curve has these first 618 
> x-coordinates as rational (or integer) squares? I would assume that 
> all x-coordinates are squares for this particular elliptic curve, but 
> that would have to be an inductive proof.
>
> Randall
>
> P.S. this also provides a list of 318 rational squares from 627^2 + 
> x[i]^2 which I also find interesting
>
> P.P.S: 627 = 3 * 11 * 19 and 627^2 = 393129