       Re: Integral points on elliptic curves

• To: mathgroup at smc.vnet.net
• Subject: [mg122319] Re: Integral points on elliptic curves
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Tue, 25 Oct 2011 06:16:27 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201110231024.GAA10524@smc.vnet.net>

```But Mathematica can easily find some solutions to this equation (and very fast):

Solve[y^3 - x^2 == 1641843 && 0 < y < 10^3, {x, y}, Integers]

{{x -> -11754, y -> 519}, {x -> -468, y -> 123},
{x -> 468, y -> 123}, {x -> 11754, y -> 519}}

The problem is to find all solutions and prove that there are no more. I don't know how to do that. It is easy, however, to prove (using the Nagell-Lutz Theorem) that the curve has no points of finite order.

Andrzej Kozlowski

On 23 Oct 2011, at 12:24, Artur wrote:

> Dear Mathematica Gurus,
> Who know that existed any Mathematica procedure (library) to finding
> integral points on elliptic curves?
> Or how to find example to e.g.
>
> FindInstance[y^3 - x^2 == 1641843, {x, y}, Integers]
>
> if FindInstance doesn't work what inspite???
>
> Unfortunatelly Wolfram Research is developing some branches of
> Mathematics in new versions of Mathematica and complete leave anothers
> (good samples are elliptic curves, Chabauty method, determining Galois
> groups of polynomials etc.).
>
> Best wishes
> Artur Jasinski
>

```

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