For the result to be an integer, all the terms under the cube roots have to be integers (because all cubes of integers are integers) and so do all the cube roots.

In particular, the rightmost must be an integer. So n = a^3, for some a.

But if a is an integer, a^3 < a^3 + a < (a+1)^3, so the cube root of a^3 + a (which is the middle term) cannot be an integer.

]]>