For two natural numbers

Question

For any two natural numbers $m$ and $n$, prove that $m^3+n^3+4$ cannot be a perfect cube.

Answer 1

Hint: Consider the equation $m^3+n^3+4=k^3$ modulo $9$. A sum of three cubes of integers is never equivalent to $\pm 4$ modulo $9$. For integers a well-known conjecture is that $x^3+y^3+z^3=n$ with $n\in \mathbb{Z}$ has a solution if and only if $n\not\equiv \pm 4\bmod 9$.