Find all of the integer solutions of $x^3y+y^3z+z^3x=0.$

Question

Using Fermat's Last Theorem, find all of the integer solutions of $x^3y+y^3z+z^3x=0.$

I try to make some substiution so as to transform the equation into a form like a fermat equation but in vain, please helps.

Answer 1

Consider a solution with $\gcd(x,y,z) = 1$. By considering primes that divide two of $x,y,z$, show that there are nonzero integers $A,B,C$ such that $x =\pm A^3 B$, $y = \pm C^3 A$, $z = \pm B^3 C$. Now what does your equation say?