If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$, is $a=b=c=0$? If $a,b,c$ are integers such that $4a^3+2b^3+c^3=6abc$ , then is it true that $a=b=c=0$ ? I was thinking of infinite descent but can't actually proceed , please help. Thanks in advance
The number $c$ would have to be even, $c = 2k$. Divide the equation by $2$ to get

$$4k^3 + 2 a^3 + b^3 = 6kab$$

This gives the infinite descent, since you can always deduce that the number with the coefficent $1$ is even.
$a=1$, $b=1$ and $c=1$ is a solution of $4a^3+2b^3+c^3=6abc$, so $a=b=c=0$ is not the only solution.