1987 CMO 第 1 题

Let $n$ be a natural number. Prove that a necessary and sufficient condition for the equation $z^{n+1}-z^n-1=0$ to have a complex root whose modulus is equal to $1$ is that $n+2$ is divisible by $6$ .

设$n$为自然数。证明方程 $z^{n+1}-z^n-1=0$ 具有模数等于 $1$ 的复数根的充分必要条件是 $n+2$ 可被 $6$ 整除。