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Re: Integral points on elliptic curves
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[1] : 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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