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Math Help - Diophantine Problems

  1. #1
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    Diophantine Problems

    I am having troubles with the following questions.

    1) Find three numbers such their sum is a square and the sum of any pair is a square?
    I set up 4 equations but couldn't solve them....
    -a+b+c=d^2
    -a+c=f^2
    -a+b=e^2
    -b+c=h^2

    how would i go about this?
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  2. #2
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    Crna Gora
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    Re: Diophantine Problems

    Quote Originally Posted by johnsy123 View Post
    I am having troubles with the following questions.

    1) Find three numbers such their sum is a square and the sum of any pair is a square?
    I set up 4 equations but couldn't solve them....
    -a+b+c=d^2
    -a+c=f^2
    -a+b=e^2
    -b+c=h^2

    how would i go about this?
    Let :

    \begin{cases}x+y+z=a^2 \\x+y=b^2 \\x+z=c^2 \\y+z=d^2\end{cases}

    then :

    2a^2=b^2+c^2+d^2

    There is an infinite number of triples (x,y,z) which represents solution .

    This Maple program calculates triples (x,y,z)

    Code:
    for a from 0 to 20 do
    for b from 0 to 20 do
    for c from 0 to 20 do
    for d from 0 to 20 do
    if 2*a^2=b^2+c^2+d^2 then
    print(solve({x+y=b^2, x+z = c^2, y+z=d^2}, [x, y, z]));
    end if;
    end do;
    end do;
    end do;
    end do;
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  3. #3
    MHF Contributor
    Joined
    Dec 2007
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    Ottawa, Canada
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    Re: Diophantine Problems

    (17,32,32) seems to be the only primitive solution with integers > 0.

    Princeps, I'd code like this (eliminate loop, no duplicates):
    for b from 0 to 20 do
    for c from b to 20 do
    for d from c to 20 do
    a = SQRT(b^2 + c^2 + d^2) / 2
    (skip if a not integer)
    Last edited by Wilmer; March 19th 2012 at 09:35 AM.
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