       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
>>
>

```

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