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>

```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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