Re: Integral points on elliptic curves

*To*: mathgroup at smc.vnet.net*Subject*: [mg122323] Re: Integral points on elliptic curves*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Tue, 25 Oct 2011 06:17:10 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201110231024.GAA10524@smc.vnet.net> <034053C0-8CD1-44D4-835E-F6EA9D174555@mimuw.edu.pl>

In fact: Baker's theorem (see Silverman's "Arithmetic of Elliptic curves" p. 261) does give an effective upper bound on the size of possible solutions x and y but in this case it is huge. Both Abs[x] and Abs[y] must be less than Exp[(10^6 1641843)^(10^6)] However even an attempt to compute this number by applying N to it produces overflow. Andrzej Kozlowski On 24 Oct 2011, at 18:03, Andrzej Kozlowski wrote: > 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>