       Re: Integral points on elliptic curves

• To: mathgroup at smc.vnet.net
• Subject: [mg122343] Re: Integral points on elliptic curves
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 26 Oct 2011 17:36:47 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201110231024.GAA10524@smc.vnet.net> <201110251016.GAA05784@smc.vnet.net> <sig.12790c7ea2.4EA6B419.1080401@csl.pl>

```Simple things like adding points on elliptic curves and related matters
are easy to implement and have, in fact, already been implemented as
demonstrations, see, e.g.

http://demonstrations.wolfram.com/RationalPointsOnAnEllipticCurve/

(I am thinking about adding something to this topic too). Implementing most of these algorithms is not very hard (there are much harder ones in Mathematica) but unless it is done by WRI they won't be efficient enough for anything but simple demonstrations. The question is, whether the subject is of sufficient general interest for WRI to devote serious resources to it.

Andrzej

On 25 Oct 2011, at 15:05, Artur wrote:

> Also method of adding torsion points can be use if we have two small
points 123, 519 we can find all other torsion points because order of
group of each elliptic curve can be determined independent method (I
have Mathematica procedure on adding points on elliptic curve). Of
course all integral points of elliptic curve are integer but not vice versa.
> If elliptic curve isn't singular number of integer points can't be bigger as 16 (is proof). Of course torsion points can be rational but never point of infinite order isn't integer.
> Theory of elliptic curves is very expanded in recent time and MAGMA contained newest algorhitms
> if we use
> Sort([ p : p in IntegralPoints(EllipticCurve([0, -1641843])) ]);
>
> on MAGMA online culculator
> http://magma.maths.usyd.edu.au/calc/
>
> results are
> [ 123, 519, 5853886516781223 ]
>
> I was think about implemnted these algorhitms in Mathematica
>
> Best wishes
> Artur Jasinski
>
> 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
>>>
>>
>>

```

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