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Math Help - Find all integer numbers

  1. #1
    Super Member dhiab's Avatar
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    Find all integer numbers

    Find all integer numbers (x,y,z) :
     <br />
x^{3}+y^{3}+z^{3}=2008<br />
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  2. #2
    Member oldguynewstudent's Avatar
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    Quote Originally Posted by dhiab View Post
    Find all integer numbers (x,y,z) :
     <br />
x^{3}+y^{3}+z^{3}=2008<br />
    The only integers that I've found to work are (12,6,4) and (10,10,2).

    So if you assign these ordered triples to x,y, and z and cycle the values you should get 12 different integer combinations of x,y, and z that will solve the equations.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by dhiab View Post
    Find all integer numbers (x,y,z) :
     <br />
x^{3}+y^{3}+z^{3}=2008<br />
    Sometimes I prefer brute force over deriving the solution in a creative way... This is one of those times.

    We must have  x,y,z<\sqrt[3]{2008}\approx12.615987<13 so there are only finitely many cases to consider.

    I wrote code to test these cases. Now it's not the most efficient code but hey it gets the job done.

    Code:
    for(int x=0;x<13;x++){
    for(int y=0;y<13;y++){
    for(int z=0;z<13;z++){
    if(x*x*x+y*y*y+z*z*z==2008){
    std::cout<<"(x,y,z)=("<<x<<","<<y<<","<<z<<")"<<std::endl;
    }}}}
    And here is the output i.e. all solutions:
    Code:
    (x,y,z)=(2,10,10)
    (x,y,z)=(4,6,12)
    (x,y,z)=(4,12,6)
    (x,y,z)=(6,4,12)
    (x,y,z)=(6,12,4)
    (x,y,z)=(10,2,10)
    (x,y,z)=(10,10,2)
    (x,y,z)=(12,4,6)
    (x,y,z)=(12,6,4)
    So as you can see there are  9 solutions.
    However if you disregard order there are  2 solutions namely  (x,y,z)=(12,6,4) \text{ and } (10,10,2) , as oldguynewstudent has already pointed out.


    Edit: I just realized you wanted your equation solved over  \mathbb{Z} , not  \mathbb{N} . This throws my idea right out the window...

    May I ask what this is for?
    Last edited by chiph588@; May 29th 2010 at 04:42 PM.
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  4. #4
    Newbie Xitami's Avatar
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    Code:
    2550 -2466 -1166
    1735 -1654  -887
    1222 -1058  -862
      361 -289  -284
       13   -5    -4
       -6   13     3
        6   12     4
        2   10    10
     -113  108    57
     -180  166   108
    -1340 1336   278
    -1152 1128   454
    -5274 5269   747
    -3770 3760   752
    -2276 2246   772
    surely more
    Last edited by Xitami; May 29th 2010 at 04:59 PM.
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Champaign, Illinois
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    Quote Originally Posted by dhiab View Post
    Find all integer numbers (x,y,z) :
     <br />
x^{3}+y^{3}+z^{3}=2008<br />
    Not sure all solutions are known by anyone...

    http://www.mathematik.uni-bielefeld....S/elkies_cubic
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