Let w = e^$\frac{2*pi*i}{3}$ = (-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+$bw^{2+cw}$)