Problem

Source: RomNO 2019, grade x, p.3

Tags: complex numbers, geometry, Complex Geometry, algebra



Find all natural numbers $ n\ge 4 $ that satisfy the property that the affixes of any nonzero pairwise distinct complex numbers $ a,b,c $ that verify the equation $$ (a-b)^n+(b-c)^n+(c-a)^n=0, $$represent the vertices of an equilateral triangle in the complex plane.