Proving that $(a+b+c)^n=a^n + b^n + c^n$

Question

Suppose that $(a+b+c)^3=a^3 + b^3 + c^3$. For what positive integer values of n is it true that $(a+b+c)^n=a^n + b^n + c^n$. Any hint will be much appreciated

Answer 1

For all odd n it is true. $(a+b+c)^3=a^3+b^3+c^3+3(a+b)(b+c)(c+a)$. Now according to the hypothesis $a+b=0 $ or $c+b=0 $ or $a+c=0 $. Say $a+b=0 $ then substituting it is true for all odd n. Similarly we can check for all other cases.