802. Iterated Composition

Let ${\mathbb{R}}^2$$\Bbb R^2$ be the set of pairs of real numbers $(x,y)$$(x, y)$. Let $\pi =3.14159\cdots$$\pi = 3.14159\cdots\ $.

Consider the function $f$$f$ from ${\mathbb{R}}^2$$\Bbb R^2$ to ${\mathbb{R}}^2$$\Bbb R^2$ defined by $f(x,y)=(x^2-x-y^2,2xy-y+\pi )$$f(x, y) = (x^2 - x - y^2, 2xy - y + \pi)$, and its $n$$n$-th iterated composition $f^{(n)}(x,y)=f(f(\cdots f(x,y)\cdots ))$$f^{(n)}(x, y) = f(f(\cdots f(x, y)\cdots))$. For example $f^{(3)}(x,y)=f(f(f(x,y)))$$f^{(3)}(x, y) = f(f(f(x, y)))$. A pair $(x,y)$$(x, y)$ is said to have period $n$$n$ if $n$$n$ is the smallest positive integer such that $f^{(n)}(x,y)=(x,y)$$f^{(n)}(x, y) = (x, y)$.

Let $P(n)$$P(n)$ denote the sum of $x$$x$-coordinates of all points having period not exceeding $n$$n$. Interestingly, $P(n)$$P(n)$ is always an integer. For example, $P(1)=2$$P(1) = 2$, $P(2)=2$$P(2) = 2$, $P(3)=4$$P(3) = 4$.

Find $P({10}^7)$$P(10^7)$ and give your answer modulo $1020340567$$1\,020\,340\,567$.

802. 迭代复合

记 ${\mathbb{R}}^2$$\Bbb R^2$ 为所有实数二元组的集合。取 $\pi =3.14159\cdots$$\pi = 3.14159\cdots\ $。

考虑一个函数 $f:{\mathbb{R}}^2\to {\mathbb{R}}^2$$f: \Bbb R^2 \rightarrow \Bbb R^2$，其对应法则为 $f(x,y)=(x^2-x-y^2,2xy-y+\pi )$$f(x, y) = (x^2 - x - y^2, 2xy - y + \pi)$。记此函数的 $n$$n$ 次复合为 $f^{(n)}(x,y)=f(f(\cdots f(x,y)\cdots ))$$f^{(n)}(x, y) = f(f(\cdots f(x, y)\cdots))$，例如 $f^{(3)}(x,y)=f(f(f(x,y)))$$f^{(3)}(x, y) = f(f(f(x, y)))$。如果某二元组 $(x,y)$$(x, y)$ 满足 $f^{(n)}(x,y)=(x,y)$$f^{(n)}(x, y) = (x, y)$，则记满足此等式的最小正整数 $n$$n$ 为该二元组的周期。

记 $P(n)$$P(n)$ 为：所有周期不超过 $n$$n$ 的二元组的第一维元素的和。有意思的是，$P(n)$$P(n)$ 一直是整数。已知：$P(1)=2$$P(1) = 2$、$P(2)=2$$P(2) = 2$、$P(3)=4$$P(3) = 4$。

求 $P({10}^7)$$P(10^7)$ 模 $1020340567$$1\,020\,340\,567$ 之值。

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