American Citizen on Thu, 24 Oct 2024 02:02:19 +0200


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


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