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Re: Integral points on elliptic curves

• To: mathgroup at smc.vnet.net
• Subject: [mg122342] Re: Integral points on elliptic curves
• From: Artur <grafix at csl.pl>
• Date: Wed, 26 Oct 2011 17:36:36 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201110231024.GAA10524@smc.vnet.net> <201110251016.GAA05784@smc.vnet.net>
• Reply-to: grafix at csl.pl

Also is possible to use LatticeReduce see
http://arxiv.org/pdf/math/0005139v1
Artur

W dniu 2011-10-25 12:16, Andrzej Kozlowski pisze:
> 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
>>
>
>

• References:
  • Integral points on elliptic curves
    • From: Artur <grafix@csl.pl>
  • Re: Integral points on elliptic curves
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>