Problem 971

Modular Polynomial Composition

Let $p$ be a prime of the form $5k-4$ and define $f_p(x) = \left(x^k+x\right) \bmod p$.

Let $C(p)$ be the number of values $0 \le x \lt p$ such that $f_p^{(m)}(x) = x$ for some positive integer $m$, that is, $x$ can be obtained by iteratively applying $f_p$ on itself starting at $x$.

For example, $C(11) = 7$, due to $x = 0, 1, 2, 3, 8, 9, 10$.

Let $S(N)$ be the sum of $C(p)$ for all primes of the form $5k-4$ not exceeding $N$. For example $S(100) = 127$.

Find $S(10^8)$.

模多项式函数复合

记$p$为形如$5k-4$的质数，并定义$f_p(x) = \left(x^k+x\right) \bmod p$。

记$C(p)$为在$0 \le x \lt p$范围内满足以下条件的整数$x$的数目：存在正整数$m$使得$f_p^{(m)}(x) = x$，或者说从$x$开始反复应用函数$f_p$最终能回到$x$。

例如，$C(11) = 7$，对应的$x = 0, 1, 2, 3, 8, 9, 10$。

记$S(N)$为所有不超过$N$且形如$5k-4$的质数$p$所对应的$C(p)$之和。例如，$S(100) = 127$。

求$S(10^8)$。