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Re: How to do quickest
*To*: mathgroup at smc.vnet.net
*Subject*: [mg123067] Re: How to do quickest
*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>
*Date*: Tue, 22 Nov 2011 05:35:50 -0500 (EST)
*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com
On 22 Nov 2011, at 10:06, Andrzej Kozlowski wrote:
>
> On 21 Nov 2011, at 10:29, Artur wrote:
>
>> Dear Mathematica Gurus,
>> How to do quickest following procedure (which is very slowly):
>>
>> qq = {}; Do[y = Round[Sqrt[x^3]];
>> If[(x^3 - y^2) != 0,
>> kk = m /. Solve[{4 m^2 + 6 m n + n^2 ==
>> x, (19 m^2 + 9 m n + n^2) Sqrt[m^2 + n^2] == y}, {m, n}][[1]];
>> ll = CoefficientList[MinimalPolynomial[kk][[1]], #1];
>> lll = Length[ll];
>> If[lll < 12, Print[{x/(x^3 - y^2)^2, kk, x, y, x^3 - y^2}];
>> If[Length[ll] == 3, Print[{kk, x, y}]]]], {x, 2, 1000000}];
>> qq
>>
>>
>> (*Best wishes Artur*)
>>
>
> I think it would be better to send not only the code but also the mathematical problem, as there may be a way to do it in a different way. Unless I am misunderstanding something, what you are trying to do is the same as this:
>
> In[31]:= Block[{y = Round[Sqrt[x^3]]},
> Reap[Table[
> If[(x^3 - y^2) != 0 && Not[IrreduciblePolynomialQ[poly]],
> Sow[{x, y}]], {x, 2, 1000000}]][[2]]] // Timing
>
> Out[31]= {721.327,{}}
>
> This ought to be a lot faster than your code, but I have not tried to run yours to the end. Also, it is possible that using the Eisenstein Test explicitly may be somewhat faster:
>
> Block[{y = Round[Sqrt[x^3]]},
> Reap[Table[
> If[x^3 - y^2 != 0 && Mod[x^6 - 2*x^3*y^2 + y^4, 4] == 0 &&
> ! IrreduciblePolynomialQ[poly], Sow[{x, y}]], {x, 2,
> 1000000}]][[2]]]
>
> {}
>
> but I forgot to use Timing and don't want to wait again, particularly that the answer is the empty set.
>
> Andrzej Kozlowski
I forgot to include the definition of poly:
Collect[poly = Eliminate[{4*m^2 + 6*m*n + n^2 == x,
(19*m^2 + 9*m*n + n^2)*Sqrt[m^2 + n^2] == y}, {n}] /. Equal -> Subtract, m]
3645*m^12 - 2916*m^10*x + m^6*(270*x^3 - 270*y^2) + x^6 -
2*x^3*y^2 + y^4
Andrzej Kozlowski
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