Let the integers be $n$, $n+1$, and $n+2$.
Notice that $n(n+1)(n+2)>$ $n^{3}$ .... (1)
Also notice that $n(n+2)<(n+1)^{2}$ $\{$since $n(n+2)=n^{2}+2n$ and $(n+1)^{2}=n^{2}+2n+1\}$
$\therefore\quad n(n+1)(n+2)<$ $(n+1)^{3}$ .... (2)
From (1) and (2), $n(n+1)(n+2)$ always lies between two consecutive perfect cubes, and so it cannot be a perfect cube.
