Show that for any positive integer $\displaystyle n$, there exist a prime number $\displaystyle p$ and another positive integer $\displaystyle m$ such that:

(1) $\displaystyle p$ is a prime numer of form $\displaystyle 6k+5$;

(2) $\displaystyle n$ is not a multiple of $\displaystyle p$;

(3) $\displaystyle n-m^3$ is a multiple of $\displaystyle p$.