       Re: Integral points on elliptic curves

• To: mathgroup at smc.vnet.net
• Subject: [mg122587] Re: Integral points on elliptic curves
• From: Artur <grafix at csl.pl>
• Date: Wed, 2 Nov 2011 06:22:28 -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> <4EAFD6E5.5090304@csl.pl>

```P.S.
Choose d^2 inspite inspite (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2)
is usefull for small d where d^2<<(4 d^2 E^2 ProductLog[-1, -(1/(2 d
E))]^2)
but if
d>9819237619956432205457343790613036173054804524960328839001963304957144\
9969310680067610622993495364457080915240696725178052317411702942199067\
4555234904932970275931679797061927319629
function d^2 is worse than (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2)
what is possible to check by
FindRoot[(4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) == d^2, {d,
10^210}, WorkingPrecision -> 1000]
but no possible to chech by Plot (becausePlot working only 10^100 range
in x axis)
Also are disagreemnet because Plot and exact numerical values of crosses
function d^y with (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2)
compare e.g.
Plot[{(4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2), d^2.05}, {d, 1,
10^100}]
where d^2.05 yet not cross with (4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2)

but croos point is range of 10^80
FindRoot[(4 d^2 E^2 ProductLog[-1, -(1/(2 d E))]^2) == d^2.05, {d,
10^210}, WorkingPrecision -> 1000]

Best wishes
Artur

W dniu 2011-11-01 12:24, Artur pisze:
> 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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