Its value isn't 4

Using Newton's sums method and let $P_n = x^n+y^n+z^n$ , where $n \in \mathbb N$ , $S_1 = x+y+z=1$ , $S_2 = xy+yz+zx$ and $S_3 = xyz$ , then we have:

\(\begin{array} {} P_1 = S_1 = 1 \\ P_2 = S_1P_1 - 2S_2 & \implies 2 = 1\cdot 1-2S_2 & \implies S_2 = \frac 12 \\ P_3 = S_1P_2 - S_2P_1+ 3S_3 & \implies 3 = 1\cdot 2 + \frac 12 \cdot 1 + 3S_3 & \implies S_3 = \frac 16 \\ P_4 = S_1P_3 - S_2P_2+ S_3P_1 & \implies P_4 = 1\cdot 3 + \frac 12 \cdot 2 + \frac 16 \cdot 1 & \implies P_4 = \frac {25}6 \end{array} \)

$\implies m + n = 25 + 6 = \boxed{31}$