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Math Help - I need help solving this 3 variable equation

  1. #1
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    I need help solving this 3 variable equation

    x^2+y^2+z^2=xyz-1

    Any help anyone? Thanks before hand.
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  2. #2
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    Re: I need help solving this 3 variable equation

    Quote Originally Posted by SMAD View Post
    x^2+y^2+z^2=xyz-1
    Any help anyone? Thanks before hand.
    It is not at all clear what you want help doing.
    When you say solve, what do you mean?
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  3. #3
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    Re: I need help solving this 3 variable equation

    By solve I mean is there a solution, which I know there isn't, but i want to know how I can show that.
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    Re: I need help solving this 3 variable equation

    Solution to what? A problem has a solution, not an equation! I suspect that this is the same problem that was posted on another board- in which it was specified that the problem was to find integer values of x, y, and z that satisfy this equation.
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    Re: I need help solving this 3 variable equation

    Quote Originally Posted by SMAD View Post
    x^2+y^2+z^2=xyz-1
    Note that x^2+y^2+z^2\ge 0 therefore xyz\ge 1.
    Clearly xyz\ne 0.
    Can you work with any of that?
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  6. #6
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    Re: I need help solving this 3 variable equation

    Quote Originally Posted by SMAD View Post
    x^2+y^2+z^2=xyz-1

    Any help anyone? Thanks before hand.
    Let us assume that you want to find integers x,y,z that satisfy the equation.

    A check on the parity of each side of the equation shows that if the equation is to work, exactly two of x,y,z must be even. Suppose WLOG that x is odd and y,z are even. Then x^2\equiv1\mod 4 (the square of any odd integer always \equiv1\mod4) and y^2 \equiv z^2 \equiv 0\mod4 (the square of any even integer is divisible by 4). Hence the LHS \equiv1\mod4. On the other hand, xyz\equiv0\mod4 and so the RHS \equiv-1\mod4. Hence the LHS can never equal the RHS, so there are no integer solutions to your equation.
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