$$x^2+2y^2+z^2=xyz$$ Diophantine Equation

[littlemathquark]

Homework Statement

How many solutions are there to the equation $$x^2+2y^2+z^2=xyz$$ where $$1\le x,y,z\le 200$$ are positive even numbers? What are the solutions?

Relevant Equations

None

I think $$x,y,z$$ must be multiple of 4 but I couldn't go on.

 

[jedishrfu]

Since all three variables are independent you can try making them the same and find a possible solution.

$$4x^2 = x^3$$

And so you can see that 4 is a solution.

Try other variations of the formula where x and z are the same or where x and y are the same.

 

[nuuskur]

If solutions must be even it's reasonable to rephrase the problem as
$$(2k)^2 + 2(2\ell)^2 + (2m)^2 = 8k\ell m \Leftrightarrow k^2 + 2\ell ^2 + m^2 = 2k\ell m,$$
where $1\leqslant k,\ell,m\leqslant 100$. The RHS is even, hence the left side must also be even. In particular, $k^2+m^2$ must be even. So $k,m$ must have the same parity.

 

[littlemathquark]

Since all three variables are independent you can try making them the same and find a possible solution.

$$4x^2 = x^3$$

And so you can see that 4 is a solution.

Try other variations of the formula where x and z are the same or where x and y are the same.

Ok, but how can I find other solutions when none of them equal each other?

 

[nuuskur]

There is no set method for solving such problems. If no solutions exist (which is not the case here), it can sometimes be shown by infinite descent. Otherwise you poke and prod and see what happens. E.g we see that $x=y=z=4$ is a solution, can other solutions exist? Try modular arithmetic as in the other post.

 

[littlemathquark]

There is no set method for solving such problems. If no solutions exist (which is not the case here), it can sometimes be shown by infinite descent. Otherwise you poke and prod and see what happens. E.g we see that $x=y=z=4$ is a solution, can other solutions exist? Try modular arithmetic as in the other post.

Thanks but modular arithmetic doesn't work on this problem, atleast for me.

 

[nuuskur]

Showing your own work would speed things along nicely.

 

[fresh_42]

The difficulty with your equation is its asymmetry. You quasi have square meters on the left and cubic meters on the right.

 

[jedishrfu]

Programming works. Ten lines of python and in seconds youll have your answer.