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Problem 854

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Pisano Periods 2

For every positive integer $n$ the Fibonacci sequence modulo $n$ is periodic. The period depends on the value of $n$. This period is called the Pisano period for $n$, often shortened to $\pi(n)$.

Define $M(p)$ as the largest integer $n$ such that $\pi(n) = p$, and define $M(p) = 1$ if there is no such $n$.

For example, there are three values of $n$ for which $\pi(n)$ equals $18$: $19, 38, 76$. Therefore $M(18) = 76$.

Let the product function $P(n)$ be:
$$P(n)=\prod_{p = 1}^{n}M(p).$$
You are given: $P(10)=264$.

Find $P(1\ 000\ 000)\bmod 1\ 234\ 567\ 891$.

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皮萨诺周期（二）

对任意正整数$n$，斐波那契数列对$n$取余的结果都是周期数列，其周期取决于$n$的取值。这一周期被称为$n$的皮萨诺周期，通常记为$\pi(n)$。

定义$M(p)$为最大的、满足$\pi(n) = p$的整数$n$。若不存在这样的$n$，定义$M(p)=1$。

例如，有三个$n$满足$\pi(n)$等于$18$，分别是$19$、$38$和$76$，因此$M(18) = 76$。

记函数$P(n)$为：
$$P(n)=\prod_{p = 1}^{n}M(p)$$
已知$P(10)=264$。

求$P(1\ 000\ 000)\bmod 1\ 234\ 567\ 891$。

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