Let w = e^(2pii/3) = (-1 + i*sqrt(3))/2 (a primitive cube root of unity).

Useful derived identities:

• (1 - w)(1 - w^2) = 3
• (1 + w)(1 + w^2) = 1 -- since 1+w = -w^2 and 1+w^2 = -w, product = w^3 = 1
• (1 - w + w^2)(1 + w - w^2) = 4 -- using 1+w+w^2=0
• a + bw + cw^2 = 0 with a,b,c real implies a = b = c (if a,b,c are distinct, then they must satisfy specific conditions)
• a^3 + b^3 = (a+b)(a+bw)(a+bw^2)
• a^3 + b^3 + c^3 - 3abc = (a+b+c)(a+bw+cw^2)(a+bw^2+cw)