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Re: How to do quickest

• To: mathgroup at smc.vnet.net
• Subject: [mg123054] Re: How to do quickest
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Tue, 22 Nov 2011 05:33:20 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201111210929.EAA14830@smc.vnet.net>
• Reply-to: drmajorbob at yahoo.com

Here's your code timed with an upper limit of 3000:

Timing[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, 3000}]]

{971/1377495072,3 Sqrt[2/5],1942,85580,52488}

{3 Sqrt[2/5],1942,85580}

{1439/117596448,Sqrt[2],2878,154396,15336}

{Sqrt[2],2878,154396}

{187.257, Null}

This is better:

Clear[x, y]
kk = m /.
    First@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}];
Timing[qq = First@Last@Reap@Do[y = x^3 // Sqrt // Round;
       (x^3 - y^2) != 0 &&

        Length@CoefficientList[MinimalPolynomial[kk][z], z] < 12 &&
        Sow@{x, y, kk, len}, {x, 2, 3000}]]

{14.0493, {{1942, 85580, -3 Sqrt[2/5], 13}, {2878, 154396, -Sqrt[2],
    13}}}

And this, even better:

Clear[x, y]
kk = m /.
    First@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}];
Timing[qq =
   First@Last@
     Reap@Do[SquareFreeQ@x && (y = x^3 // Sqrt // Round; True) &&

          Length@CoefficientList[MinimalPolynomial[kk][z], z] < 12 &&
        Sow@{x, y, kk, len}, {x, 2, 3000}]]

{8.39548, {{1942, 85580, -3 Sqrt[2/5], 13}, {2878, 154396, -Sqrt[2],
    13}}}

All this is VERY slow nonetheless. Maybe there's another way to  
characterize the problem?

Bobby

On Mon, 21 Nov 2011 03:29:38 -0600, Artur <grafix at csl.pl> 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*)
>


-- 
DrMajorBob at yahoo.com

• References:
  • How to do quickest
    • From: Artur <grafix@csl.pl>