Let $b\ge 2$ be an integer, and write the base $b$ expansion of any non-negative integer $n$ as $n=x_0+x_{1}b+\cdots +x_d{b}^d$, with $x_d>0$ and $0\le x_i<b$ for $i=0,\dots ,d$. Let $\varphi (x)$ denote an integer polynomial such that $\varphi (n)>0$ for all $n>0$. Consider the map $S_{\varphi ,b}:{\mathbb{Z}}_{\ge 0}\to {\mathbb{Z}}_{\ge 0}$, with $S_{\varphi ,b}(n):=\varphi (x_0)+\cdots +\varphi (x_d)$. It is known that the orbit set $\{n,S_{\varphi ,b}(n),S_{\varphi ,b}(S_{\varphi ,b}(n)),\dots ,\}$ is finite for all $n>0$. Each orbit contains a finite cycle, and for a given $b$, the union of such cycles over all orbit sets is finite.

Fix now an integer $\ell \ge 1$ and let $\varphi (x)=x^2$. We show that the set of bases $b\ge 2$ which have at least one cycle of length $\ell$ always contains an arithmetic progression and thus has positive lower density. We also show that a 1978 conjecture of Hasse and Prichett on the set of bases with exactly two cycles needs to be modified, raising the possibility that this set might not be finite.

让 $乙\ge 2$ 是一个整数，并写下基数 $乙$ 任何非负整数的展开 $n$ 作为 $n=X_0+X_1乙+\cdots +X_d{乙}^d$， 和 $X_d>0$ 和 $0\le X_{\mathrm{一世}}<乙$ 为了 $\mathrm{一世}=0,\dots ,d$. 让$\phi (X)$ 表示一个整数多项式，使得 $\phi (n)>0$ 对所有人 $n>0$. 考虑地图${秒}_{\phi ,乙}：{\mathbb{Z}}_{\ge 0}\to {\mathbb{Z}}_{\ge 0}$， 和 ${秒}_{\phi ,乙}(n)：=\phi (X_0)+\cdots +\phi (X_d)$. 众所周知，轨道集$\{n,{秒}_{\phi ,乙}(n),{秒}_{\phi ,乙}({秒}_{\phi ,乙}(n)),\dots ,\}$ 对所有人都是有限的 $n>0$. 每个轨道包含一个有限周期，对于给定的$乙$，所有轨道集上的这些循环的并集是有限的。

现在修复一个整数 $\ell \ge 1$ 然后让 $\phi (X)=X^2$. 我们证明了基集$乙\ge 2$ 至少有一个周期的长度 $\ell$总是包含一个等差数列，因此具有正的较低密度。我们还表明，Hasse 和 Prichett 在 1978 年关于恰好有两个循环的基集的猜想需要修改，这增加了该集可能不是有限集的可能性。