I need help solving this 3 variable equation

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May 18th 2012, 03:02 AM
SMAD

I need help solving this 3 variable equation

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

Any help anyone? Thanks before hand.
May 18th 2012, 04:18 AM
Plato

Re: I need help solving this 3 variable equation

Quote:

Originally Posted by SMAD

$x^2+y^2+z^2=xyz-1$
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?
May 18th 2012, 09:05 AM
SMAD

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.
May 18th 2012, 09:11 AM
HallsofIvy

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.
May 18th 2012, 09:17 AM
Plato

Re: I need help solving this 3 variable equation

Quote:

Originally Posted by SMAD

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?
May 18th 2012, 09:40 AM
Sylvia104

Re: I need help solving this 3 variable equation

Quote:

Originally Posted by SMAD

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.