Prove that the equation $x^4+y^4+z^4-2x^2y^2-2y^2z^2-2z^2x^2=24$ has no solutions in integers.
When $x,y,z$ are all even no solution exists as the LHS is divisible by 16 whereas RHS is not.
So now, remains two cases:
i) One of $x,y,z$ is odd and two are even.
ii) Two of $x,y,z$ are odd and one is even.
Case i) -say $x$ is odd. LHS stands as $x^4+16k_1+16k_2-8k_3x^2-32k_4-8k_4x^2$. We are now to consider, $x^4-2x^2y^2-2x^2z^2$ as other elements are divisible by $16$. If we can prove that this part is divisible by $16$ then we are done but I cannot do that.
As for Case ii) I could not think of how to show it divisible by $16$. So, I think I need to find out something else that works which I could not. Please help.