Re: question on diophantine equations in Mathematica

*To*: mathgroup at smc.vnet.net*Subject*: [mg115502] Re: question on diophantine equations in Mathematica*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Thu, 13 Jan 2011 03:24:39 -0500 (EST)

I am using Mathematica 8 on 2.66 Ghz Mac Book Pro with 8 gigabytes of Ram. The time is measured in seconds. With Mathematica 8 you can also get the same answer with Solve: Timing[ Solve[x^10 + y^10 + z^10 == t^2 && 0 <= x && x == y && y <= z && 1 <= t <= 90000, {x, y, z, t}, Integers]] {1.01969,{{x->0,y->0,z->1,t->1},{x->0,y->0,z->2,t->32},{x->0,y->0,z->3,t->243},{x->0,y->0,z->4,t->1024},{x->0,y->0,z->5,t->3125},{x->0,y->0,z->6,t->7776},{x->0,y->0,z->7,t->16807},{x->0,y->0,z->8,t->32768},{x->0,y->0,z->9,t->59049}}} I think for very large numbers PowersRepresentations will give you more satisfactory answers. For example, compare the output Solve[x^10 + y^10 + z^10 == 10^20 && 0 <= x && x <= y && y <= z, {x, y, z, t}] with PowersRepresentations[10^21, 3, 10^20] {} Andrzej Kozlowski On 12 Jan 2011, at 13:16, Ivan Smirnov wrote: > Hello, Andrzej. > Many thanks for reply. > What PC do you use (OS, CPU & RAM) and how many minutes did it take to compute, what is 0.872...? > What is the upper margin for t which can cause overflow? > Do you have any other ideas how to increase performance for my task? > Will be very glad for help > > 2011/1/12 Andrzej Kozlowski <akoz at mimuw.edu.pl> > This seems to show that there are only trivial solutions for 1<=t<==90000 > > Timing[ > Select[Table[ > PowersRepresentations[t^2, 3, 10], {t, 1, 90000}], #1 != {} & ]] > > {0.8722430000000259, {{{0, 0, 1}}, {{0, 0, 2}}, {{0, 0, > 3}}, {{0, 0, 4}}, {{0, 0, 5}}, {{0, 0, 6}}, {{0, 0, 7}}, > {{0, 0, 8}}, {{0, 0, 9}}}} > > The algorithm basically uses "brute force" so you will start getting overflows for very large t. > > Andrzej Kozlowski > > > > On 12 Jan 2011, at 01:25, Ivan Smirnov wrote: > > > Hi all, > > I've installed trial of Mathematica 8. > > I would like to search for possible solutions of diophantine equation > > x^10+y^10+z^10==t^2. > > How to do this efficiently? > > FindInstance seems to be VERY slow! And indeed it doesn't always find every > > solution of diophantine equations. For example I've tried it with > > x^4+y^4+z^4==t^4 and it didn't find anything (but there are solutions!). > > And Solve command just don't want to search! With some seconds it gives > > During evaluation of In[1]:= Solve::svars: Equations may not give solutions > > for all "solve" variables. >> > > I will be very glad if someone make INDEED FAST algorithm for searching. > > > > Ivan > >