Re: How to do quickest
- To: mathgroup at smc.vnet.net
- Subject: [mg123087] Re: How to do quickest
- From: Artur <grafix at csl.pl>
- Date: Wed, 23 Nov 2011 07:04:46 -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>
- Reply-to: grafix at csl.pl
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 answers: > > 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 > > >
- References:
- How to do quickest
- From: Artur <grafix@csl.pl>
- Re: How to do quickest
- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
- How to do quickest