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Re: Integral points on elliptic curves

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122573] Re: Integral points on elliptic curves
  • From: Artur <grafix at csl.pl>
  • Date: Wed, 2 Nov 2011 06:19:55 -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>
  • Reply-to: grafix at csl.pl

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
>>>
>
>



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