<s>[INST] Number of solutions in degree four Find number of postive integra solutions of the equation
$ x^4+ 4y^4 + 16z^4 +64= 32xyz$.
I could just proceed till that x cant be odd.
 [/INST] There are no integer solutions.
As you remark, $x$ must be even. Write $x=2x_1$.  Then we see $$16x_1^4+4y^4+16z^4+64=64x_1yz\implies 16\,|\,4y^4\implies y=2y_1$$
Continuing we get $$4x_1^4+16y_1^4+4z^4+16=16x_1y_1z\implies x_1^4+4y_1^4+z^4+4=4x_1y_1z$$
Thus, $x_1\equiv z\pmod 2$.  If they were both odd, we'd have $x_1^4,z^4\equiv 1 \pmod 4$ in which case out last equation becomes $2\equiv 0 \pmod 4$, a contradiction.  Thus they are both even. Write $x_1=2x_2, z=2z_1$.  We have $$16x_2^4+4y_1^4+16z_1^4+4=16x_2y_1z_1\implies 4x_2^4+y_1^4+4z_1^4+1=4x_2y_1z_1$$
Thus $y_1$ must be odd.  But in that case $y_1^4\equiv 1\pmod 4$ in which case our last equation becomes $2\equiv 0 \pmod 4$, a contradiction, and we are done.
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