Re: Integral points on elliptic curves

• To: mathgroup at smc.vnet.net
• Subject: [mg122472] Re: Integral points on elliptic curves
• From: Costa Bravo <q13a27tt at aol.com>
• Date: Sat, 29 Oct 2011 07:12:50 -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> <j8dthn\$kga\$1@smc.vnet.net>

```Andrzej Kozlowski wrote:

>> 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

O , You have y<10^3  I have y <= 10 000 !!

Solve[y^3 - x^2 == 1641843&&  0<  x&&  0<  y<  10^4, {x, y},
Integers] // Timing

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

This is hardly impressive ;)

If in your algorithm, we take y< 2*10^4 -> Solve::svars !

--
Costa

```

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