The minimum value of `|a+bomega+comega^(2)|`, where a,b,c are all not equal integers and `omega (ne 1)` is a cube root of unity, is

Explanation

Let `x = |a + b omega + comega^(2)|`
` Rightarrow x^(2) = |a + bomega +comega^(2)| = ( a+bomega + comega^(2) + comega) (a + bbaromega + cbaromega^(2)) [a ,b ,c " are integer " , bara =a ]`
`= (a + bomega+comega^(2))(a+bomega^(2) +comega), baromega=omega^(2)and baromega^(2) =omega`
`Rightarrow x^(2) =a^(2)+b^(2) +c^(2) -ab-ca=1/2[(a-b)^(2) +(b-c)^(2) +(c -a)^(2)]`
` Rightarrow x^(2) =a^(2) +b^(2)+c^(2) -ab -bc-ca = 1/2 [(a-b)^(2) +(b -c)^(2) +(c-a)^(2)]`
`Rightarrow x^(2)=1/2[(a-b)^(2) +(b-c)^(2) +(c-a)^(2)]`
Since, a,b,c are intergers but not all simultaneously equal hence we may assume two of them equal , say b =c but ` a ne b and a ne c`.
` Rightarrow ( b-c)^(2) =0, (a-b)^(2) ge 1 and (a-c)^(2) ge 1,` as the difference between two consecutive intergers is `+-1` .
` Rightarrow x^(2) ge 1/2 [ 1 + 0 +1] =1`
` |x| ge 1 or |x| le -1 " But " |x| ne -ve`
` Rightarrow min |x| =1`