Find all the integral solutions of x(cube) = 6y(cube) + 20z(cube)

Since there are three variables x,y,z.Hence we need three equations to solve them

I have asked for integral solutions not its exact value. If it was to be exact value then three equations were required but I need integral solutions.

Its evident that x is even. This means LHS is divisible by 8

Now if y is odd and z even, then RHS leaves remainder 6 on division by 8

If y is even and z is odd, then RHS leaves remainder 4

If both y and z are odd, then RHS leaves remainder 2.

Hence y and z are both even

Let x=2p, y=2q, z=2r. Then after cancellation we have 

That means if (x,y,z) is a solution so is (p,q,r).

But this process can be continued indefinitely, which is absurd (Infinite Descent)

Hence no solutions are possible