If and are distinct integers in a geometric progression which satisfy the equation above where is a non-real cube root of unity, what is the least possible value of ?
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Let a, b, c respectively be a, ar, ar^2. The given expression simplifies to a^3((1+ rw+ r^2w^2)^3+ (1+ rw^2+ r^2w)^3)=0. Applying the formula of x^3+ y^3. We get; 2-r- r^2=0. r=1 or -2. r can not be 1 as a, b, c are distinct. Hence sum reduces to a(1+2+7) =7a. Minmum value 7.