Re: How to do quickest

• To: mathgroup at smc.vnet.net
• Subject: [mg123090] Re: How to do quickest
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 23 Nov 2011 07:05:21 -0500 (EST)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <201111210929.EAA14830@smc.vnet.net> <8FECCEBE-CBFE-4B7C-87A5-4856C65C2DB4@mimuw.edu.pl> <7F85110E-2F9F-4B87-8F98-C7EEF63DEB4D@mimuw.edu.pl> <201111221223.HAA00196@smc.vnet.net> <sig.13073ac2f6.4ECBD594.7060709@csl.pl> <5E76C41D-6FD7-4FFD-B1D2-979439064056@mimuw.edu.pl>

```I think the correct use of Eisenstein's criterion is this:

Block[{y = Round[Sqrt[x^3]]},
Reap[Table[
If[x^3 - y^2 != 0 &&
Not[Mod[x^6 - 2*x^3*y^2 + y^4, 2] == 0 &&
Mod[x^6 - 2*x^3*y^2 + y^4, 4] != 0] && !
IrreduciblePolynomialQ[poly], Sow[{x, y}]], {x, 2,
1000000}]][[2]]] // Timing

This is because 3645 is not divisible by 2, but  2916 and 270 both are.
So if the term x^6-2 x^3 y^2+y^4 is divisible by 2 but not divisible by
4, the polynomial is irreducible and there is no need to test it
further. Only when this isn't the case, we need to use
IrreduciblePolynomialQ.

I am not sure if adding this will speed up the code. I only have one
copy of Mathematica and I need to use it so I can't afford the time to
run these programs now, when I am a hurry.

Andrzej

On 22 Nov 2011, at 19:30, Andrzej Kozlowski 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 =
>>>
>>> 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
>>>
>>>
>>>
>>>
>

```

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