[seqfan] Re: Help needed

[Max Alekseyev]

On Fri, Feb 5, 2010 at 7:30 AM, Artur <grafix at csl.pl> wrote:
> Dear Sequans:
> We are looking for such x that
> (37 + 2 x + x^2)/(-3 + 4 x) is perfect square
>
> How to proof that set x={2,7,32} is finite or find other x ?

Let (37 + 2 x + x^2)/(-3 + 4 x) = y^2 then
(37 + 2 x + x^2) - y^2 * (-3 + 4 x) = 0.

This is an elliptic curve and its Weierstrass form is:

v^2 = u^3+1679/3*u-73262/27+

under the transformation:

x = (9*u+1833)/(6*u-28)/2
y = 3*v/(12*u-56)

It is easy to check (e.g., in PARI/GP) that the rank of this curve is
zero, while the torsion subgroup is of order 8 and generated by a
point [u,v]=[29/3, 60].
Therefore, all rational points on this curve are:
O, [29/3, 60], [89/3, -200], [389/3, 1500], [14/3, 0], [389/3, -1500],
[89/3, 200], [29/3, -60]

They correspond to the following points on the original curve:
[32, 3], [7, -2], [2, 3], [2, -3], [7, 2], [32, -3]

Regards,
Max