On Dec 7, 7:21 pm, John Cremona <john.crem...@gmail.com> wrote:
> PS Your second example is a Weierstrass model but not integral:
>
> sage: E = EllipticCurve([0,0,0,0,-81/4])
> sage: E.integral_points()
> ---------------------------------------------------------------------------
> ...
> ValueError: integral_points() can only be called on an integral model
>
> But you can do this:
>
> sage: E1 = E.integral_model(); E1
> Elliptic Curve defined by y^2 = x^3 - 1296 over Rational Field
> sage: E1.integral_points()
> [(193 : 2681 : 1)]
>
> and also this:
>
> sage: E1.S_integral_points([2])
> [(193 : 2681 : 1)]
>
> which I think shows taht the only integral (or {2}-integral) solution
> to your equation is x=193, y=2681. Does that help?
I think it might help but I don't know the theory of algebraic curves
so much that I can give a valid proof of the integer points. I found
the problem at http://www.mathlinks.ro/viewtopic.php?t=150659 where
one solution was given but I was wondering if there is some other way
to find the integer points. However, I haven't found any proof which
uses advanced methods.
Jaakko
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