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>