Each of x and y is a positive integer.

Can each of x^3+y and y^3+x be a *perfect cube?*.

Give reasons for your answer.

No, they cannot both be a perfect cube.

Without loss of generality, assume that y >= x.

Then y^3 + x cannot be a perfect cube, because it is greater than y^3 but less than (y+1)^3.

To see this, note that (y+1)^3 is y^3 + 3y^2 + 3y + 1, which is obviously greater than y^3 + x whenever y >= x.

q.e.d.