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

9 years ago

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.

9 years ago 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