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Re: question on diophantine equations in Mathematica
*To*: mathgroup at smc.vnet.net
*Subject*: [mg115544] Re: question on diophantine equations in Mathematica
*From*: Adam Strzebonski <adams at wolfram.com>
*Date*: Fri, 14 Jan 2011 06:17:38 -0500 (EST)
For Diophantine equations Solve and Reduce use the same code, so there
should be no speed difference.
Best Regards,
Adam Strzebonski
Wolfram Research
Ivan Smirnov wrote:
> Am I right that with such constraints Reduce now is many faster than Solve?
>
> 2011/1/14 Adam Strzebonski <adams at wolfram.com <mailto:adams at wolfram.com>>
>
> Reduce does a case-by-case search here, but it chooses the variables
> with smallest ranges of values for the search.
>
> In[1]:= form = x^10 + y^10 + z^10 == t^2 && x >= 0 && y >= 1 &&
> x <= y && y <= z && 1 <= t <= 250000;
>
> In[2]:= vars = {x, y, z, t};
>
> In[3]:= {Ceiling[MinValue[#, form, vars]],
> Floor[MaxValue[#, form, vars]]}&/@vars
>
> Out[3]= {{0, 10}, {1, 11}, {1, 12}, {2, 250000}}
>
> Reduce does case-by-case search for the first 3 variables,
> so the problem reduces to solving
>
> In[4]:= 11 11 12
> Out[4]= 1452
>
> univariate quadratic equations. The most time-consuming part
> here is computing the MinValue/MaxValue (8 CAD problems with
> 4 variables).
>
> This method will do exhaustive search on {x, y, z} as long as
> the number of possible {x, y, z} values does not exceed 10000.
>
> If you want it to do larger searches you need to change
> the second value of the system option
>
> In[5]:= "ExhaustiveSearchMaxPoints"/.("ReduceOptions"/.SystemOptions[])
> Out[5]= {1000, 10000}
>
> This increases the maximum number of search points to 10^6.
>
> In[6]:= SetSystemOptions["ReduceOptions"->{"ExhaustiveSearchMaxPoints"->
> {1000, 10^6}}];
>
> This proves that the problem has no solutions with t <= 10^10
> (a search over 828630 cases).
>
> In[7]:= Reduce[x^10 + y^10 + z^10 == t^2 && 0 <= x && 0 < y &&
> x <= y && y <= z && 1 <= t <= 10^10, {x, y, z, t}, Integers] // Timing
>
> Out[7]= {269.967, False}
>
>
> Best Regards,
>
> Adam Strzebonski
> Wolfram Research
>
>
>
> Andrzej Kozlowski wrote:
>
> Actually Solve (and Reduce) are remarkably efficient at solving
> this problem. It is better to reformulate it by requiring that
> the two smallest values be larger than 0. This eliminates all
> trivial solutions. Solve (and Reduce) then work remarkably fast
> for very large numbers t, e.g.
>
> Solve[
> x^10 + y^10 + z^10 == t^2 && 0 <= x && 0 < y && x <= y && y <=
> z && 1 <= t <= 250000, {x, y, z, t}, Integers] // Timing
>
> {1.80372,{}}
>
> Now look at this:
>
>
> In[1]:= Solve[
> x^10 + y^10 + z^10 == t^2 && 0 <= x && 0 < y && x <= y && y <=
> z && 1 <= t <= 350000, {x, y, z, t}, Integers] // Timing
>
> Out[1]= {1.90608,{}}
>
> This is so fast that it almost excludes the possibility of "case
> by case" verification. Moreover, solving the problem for t=
> 350,000 took only slightly longer than for t= 250,000.
> If this is indeed a valid proof (and I think it is - Reduce
> gives the same answer) then it looks like Solve is really using
> some knowledge of Diophantine equations to solve this. It would
> be really interesting to know what is going on. I almost
> inclined to believe that Solve is able to prove that there are
> no solutions for all t, but running it without a bound on t
> produces a useless (in this context) "conditional" answer:
>
> Solve[x^10+y^10+z^10==t^2&&0<=x&&0<y&&x<=y&&y<=z,{x,y,z,t},Integers]
> (Output deleted).
>
> I do hope we get some insight into what Solve is doing. It is
> beginning to look fascinating, although I am probably missing
> something simple...
>
> Andrzej Kozlowski
>
> On 12 Jan 2011, at 19:41, Andrzej Kozlowski wrote:
>
> Yes, but I think I unintentionally deceived you (and
> myself). Mathematica caches its results and recall that I
> tried solving this several times with different numbers
> before I run Timing. mathematica obviously remembered all
> the answers and when I tried measuring the time taken it was
> amazingly fast. I should have found it suspicious but as I
> was busy with other things I did not notice anything.
>
> Now that I tried again with a fresh kernel the results are
> much less impressive: in fact much closer to yours.
> But note that now it is clear that Solve is very much faster
> than PowersRepresentatins (it looked the other way round
> before). In fact Solve deals with this impressively fast:
>
> Timing[
> Select[Table[
> PowersRepresentations[t^2, 3, 10], {t, 1, 90000}], #1 != {}
> & ]]
>
> {641.343049, {{{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}}}}
>
>
> Timing[
> Solve[x^10 + y^10 + z^10 == t^2 && 0 <= x && x <= y && y <=
> z && 1 <= t <= 90000, {x, y, z, t}, Integers]]
>
> {1.161798, {{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}}}
>
> This suggests strongly that you should in fact use Solve.
> However, you should not try to test for too large a group of
> solutions at a time. For example, you can get the next
> 10,000 quickly:
>
> Timing[
> Solve[x^10 + y^10 + z^10 == t^2 && 0 <= x && x <= y && y <=
> z && 90000 <= t <= 100000, {x, y, z, t}, Integers]]
>
> {1.48964,{{x->0,y->0,z->10,t->100000}}}
>
> But the time for this 10,000 is longer than for the previous
> 90,000!
>
> With best regards
>
> Andrzej
>
>
>
> On 12 Jan 2011, at 16:03, Ivan Smirnov wrote:
>
> Oh, it's very cool computer. What model of CPU, Intel or
> AMD?
> I just thought that time was in seconds, but surprised
> that it took so few time and asked.
> I have only 1.6 Ghz Acer (Intel T5500) with 1 GB Ram, so
> for 1..90000 it took 1034 seconds to compute...
>
> 2011/1/12 Andrzej Kozlowski <akoz at mimuw.edu.pl
> <mailto:akoz at mimuw.edu.pl>>
> 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
> <mailto: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
>
>
>
>
>
>
>
>
>
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