Let the integers be n, n+1, and n+2.
Notice that n(n+1)(n+2)> n3 .... (1)
Also notice that n(n+2)<(n+1)2 {since n(n+2)=n2+2n and (n+1)2=n2+2n+1}
∴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.