Re: How to do quickest

• To: mathgroup at smc.vnet.net
• Subject: [mg123126] Re: How to do quickest
• From: DrMajorBob <btreat1 at austin.rr.com>
• Date: Thu, 24 Nov 2011 06:55:28 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201111210929.EAA14830@smc.vnet.net>

```I'm getting all these REPEATED, 2 days after the first round.

Great.

Bobby

On Wed, 23 Nov 2011 06:05:09 -0600, Andrzej Kozlowski <akoz at mimuw.edu.pl>
wrote:

> My memory of Eisenstein's criterion was wrong (also, I was too much in a
> hurry to look it up).  Rather than correcting it I got rid of it
> altogether since I think Mathematica probably uses it anyway. I then
> get:
>
> In[1]:= 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]
>
> Out[1]= 3645 m^12-2916 m^10 x+m^6 (270 x^3-270 y^2)+x^6-2 x^3 y^2+y^4
>
> In[2]:= 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[2]= {1089.54,({1942,85580}	{2878,154396}	{3862,240004}
> {6100,476425}	{8380,767125}	{11512,1235168}	{15448,1920032}
> {18694,2555956}	{31228,5518439}	{93844,28748141}
> {111382,37172564}	{117118,40080716}	{129910,46823500}
> {143950,54615700}	{186145,80311375}	{210025,96251275}
> {375376,229985128}	{445528,297380512}	{468472,320645728}
> {575800,436925600}	{844596,776199807}	{950026,925983476}
>
> )}
>
> This gets all the numbers but is much slower (I guess it will be better
> to add the Eisenstein criterion after all, but of course, in correct
> form).
>
> Andrzej Kozlowski
>
>
>
>
> On 22 Nov 2011, at 18:02, Artur wrote:
>
>> Dear Andrzej,
>> Your procedure omiited some points
>> 6100
>> 8380 Best wishes
>> Artur
>>
>> W dniu 2011-11-22 13:23, Andrzej Kozlowski pisze:
>>> On 22 Nov 2011, at 10:07, Andrzej Kozlowski wrote:
>>>
>>>
>>>> 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
>>>>
>>>
>>> Strange but I run this code with a fresh kernel and got the following
>>>
>>> In[1]:=
> 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]
>>> Out[1]= 3645 m^12-2916 m^10 x+m^6 (270 x^3-270 y^2)+x^6-2 x^3 y^2+y^4
>>>
>>> In[2]:=
> 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]]]//Timing
>>>
>>> Out[2]=
>>> {766.05,({1942,85580}	{2878,154396}	{3862,240004}	{11512,1235168}	{15448,1920032}	{18694,2555956}	{111382,37172564}	{117118,40080716}	{129910,46823500}	{143950,54615700}	{186145,80311375}	{210025,96251275}	{375376,229985128}	{445528,297380512}	{468472,320645728}	{575800,436925600}	{950026,925983476}
>>>
>>> )}
>>>
>>>
>>> I tested the first one and it does seem to be a solution to your
>>> problem.
>>>
>>> {x, y} = {950026, 925983476};
>>>
>>> y == Round[Sqrt[x^3]]
>>>
>>> True
>>>
>>> x^3 - y^2 != 0
>>>
>>> True
>>>
>>> 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]]
>>>
>>> Out[12]= -Sqrt[-(198/5)-(44 I Sqrt[11])/5]
>>>
>>> ll = CoefficientList[MinimalPolynomial[kk][[1]], #1];
>>>
>>> Length[ll]
>>>
>>> 5
>>>
>>> I don't know why I got no answers the first time round, perhaps one of
>>> the variables had values assigned.
>>>
>>> Andrzej
>>>
>>>
>>>
>>>
>
>

--
DrMajorBob at yahoo.com

```

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