Question

If a,b,c are integers, not all simultaneously equal and w is cube root of unity (`w != 1)`, then minimum value of ` |a + bw + c w^2|`

Answer 1

`Z=a+bw+cw^2`
`Z=a+bw^2+cw`
`|z^2|=Z*underlineZ=(a+bw+cw^2)(a+bw^2+cw)`
`=a^2+b^2+c^2+abw^2+acw+acw^2+bcw+bcw^2+abw`
`|Z|^2=a^2+b^2+c^2-ab-bc-ac`
`|z|^2=((a-b)^2+(b-c)^2+(a-c^2))/2`
`|z|=sqrt((a-b)^2+(b-c)^2+(a-c^2))/2`
a=b=n,c=n+1,n-1
`|Z_(min)|=sqrt((0+1+1)/2)=1`
`Ifa!=b!=c`
`a=n,b=n-1,c=n+1`
`Z_(minn)=sqrt((1^2+1^2+2^2)/2)=sqrt3`