If a,b,c are integers, no two of them are equal then minimum value of $|a+bw+c{w}^2|$ is ( where w is complex cube root of unity)

$$\begin{gathered} {|a+bw+c{w}^2|}^2 \\ =(a+bw+c{w}^2)(a+b{w}^2+cw) \\ =a^2+b^2+c^2-ab-bc-ca \\ =\frac{1}{2}[{(a-b)}^2+{(b-c)}^2+{(c-a)}^2] \end{gathered}$$

which is least if a,b,c are consecutive integers
$\therefore$ minimum value  $=\sqrt{3}$