What is minimum value of |a+bw+cw^2|? whrere w is cube root of unity?

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The discussion centers on finding the minimum value of the expression |a + bw + cw^2|, where w is a cube root of unity and a, b, c are integers that are not all equal. It is established that |a + bw + cw^2| can be bounded by |a| + |bw| + |cw^2|, leading to the conclusion that this sum is at least a + b + c. However, there is confusion regarding whether this minimum value can be achieved, particularly questioning if |w^2| equals 1. The participants explore the implications of these conditions on the minimum value of the expression. Ultimately, the discussion highlights the complexities involved in determining the minimum value given the constraints on a, b, and c.
vkash
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a,b,c are integers not all equal and w is the cube root of unity then minimum value of |a+bw+cw2|(w is not equals one).

My answer
|a+bw+cw2|<=|a|+|bw|+|cw2|
|a|+|bw|+|cw2|=a+b+c.
so at lest one value of |a+bw+cw2| will smaller than the minimum value of a+b+c. for integers this minimum value is smallest integers you can think of.
But that's wrong.?why?
 
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Is |w2| = 1?
 

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