Suppose $\gcd(a,b,c)=1$. Then is $a^n+b^n+c^n$ divisible by $a+b+c$?

Question

I tried some examples and they all worked, but I have no idea how to factorize $a^n+b^n+c^n$ into the product of $a+b+c$ and something else.

Could anyone tell me whether this result is true or not? And, if it is true, how to prove it?

Answer 1

There’s a well-known [classical] recursion:

$a^n+b^n+c^n = (a+b+c)(a^{n-1}+b^{n-1}+c^{n-1})-(ab+ac+bc)(a^{n-2}+b^{n-2}+c^{n-2})+abc(a^{n-3}+b^{n-3}+c^{n-3}).$

I don’t believe there’s any factorization with only $a+b+c$.