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Re: Integral points on elliptic curves

• To: mathgroup at smc.vnet.net
• Subject: [mg122434] Re: Integral points on elliptic curves
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Fri, 28 Oct 2011 05:36:27 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <j80q8p$adb$1@smc.vnet.net> <j862sg$5v7$1@smc.vnet.net> <201110262140.RAA00128@smc.vnet.net>

On 26 Oct 2011, at 23:40, Costa Bravo wrote:

>  Emu write:
>>>
>>> FindInstance[y^3 - x^2 == 1641843, {x, y}, Integers]
>>>
>>> if FindInstance doesn't work what inspite???
>>>
>
>> How about something as naive as
>>
>> In[31]:= Cases[Join @@ Table[{x, y, y^3 - x^2 == 1641843}, {x, 1,
>> 1000}, {y, 1, 1000}], {__, True}]
>> Out[31]= {{468, 123, True}}
>
> In[51]:= AbsoluteTiming[
>  Cases[Join @@  Table[{x, y, y^3 - x^2 == 1641843}, {x, 1, 1000}, {y, 1,
>      1000}], {__, True}]]
>
> Out[51]= {4.6718750, {{468, 123, True}}}
>
> better
>
> In[60]:= k = 1641843;
> AbsoluteTiming[y0 = Ceiling[k^(1/3)];
>  Do[If[FractionalPart[Sqrt[y^3 - k]] == 0,Print[Sqrt[y^3 - k], "  ", y]], {y, y0, 10000}]]
>
>   468  123
> 11754  519
>
> Out[61]= {3.4375000, Null}
>
> --
>  Costa
>

This is hardly impressive when compared with:

 Solve[y^3 - x^2 == 1641843 && 0 < x && 0 < y < 10^3, {x, y},
  Integers] // Timing

{0.383947,{{x->468,y->123},{x->11754,y->519}}}

Andrzej Kozlowski

• References:
  • Re: Integral points on elliptic curves
    • From: Costa Bravo <q13a27tt@aol.com>