Re: Integral points on elliptic curves

*To*: mathgroup at smc.vnet.net*Subject*: [mg122587] Re: Integral points on elliptic curves*From*: Artur <grafix at csl.pl>*Date*: Wed, 2 Nov 2011 06:22:28 -0500 (EST)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*References*: <201110231024.GAA10524@smc.vnet.net> <034053C0-8CD1-44D4-835E-F6EA9D174555@mimuw.edu.pl> <201110251017.GAA05822@smc.vnet.net> <4EAFD6E5.5090304@csl.pl>*Reply-to*: grafix at csl.pl

P.S. Choose d^2 inspite inspite (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) is usefull for small d where d^2<<(4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) but if d>9819237619956432205457343790613036173054804524960328839001963304957144\ 9969310680067610622993495364457080915240696725178052317411702942199067\ 4555234904932970275931679797061927319629 function d^2 is worse than (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) what is possible to check by FindRoot[(4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) == d^2, {d, 10^210}, WorkingPrecision -> 1000] but no possible to chech by Plot (becausePlot working only 10^100 range in x axis) Also are disagreemnet because Plot and exact numerical values of crosses function d^y with (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) compare e.g. Plot[{(4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2), d^2.05}, {d, 1, 10^100}] where d^2.05 yet not cross with (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) but croos point is range of 10^80 FindRoot[(4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) == d^2.05, {d, 10^210}, WorkingPrecision -> 1000] Best wishes Artur W dniu 2011-11-01 12:24, Artur pisze: > Baker theory (not proof) isn't constructive from algorhitmic point of > view. Much better is Hall conjecture. > If we take additional condition that Hall conjecture is true in such > case Abs[x] is less than (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) > which isn't overflow for Mathematica for resonable d values d=x^3-y^2 > > Some programmes working that these limits for x are much more > decreased but exceptions are stored in separate base of points with > big canonical high. > If we take very sharp limit x <d^2 only 25 exceptions is know for > d=x^3-y^2 > see e.g. > http://www.math.harvard.edu/~elkies/hall.html > > Best wishes > Artur > > W dniu 2011-10-25 12:17, Andrzej Kozlowski pisze: >> 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 >>>> >> >>