  Let $b \geq 2$ be an integer, and write the base $b$ expansion of any
non-negative integer $n$ as $n=x_0+x_1b+\dots+ x_{d}b^{d}$, with $x_d>0$ and $
0 \leq x_i < b$ for $i=0,\dots,d$. Let $\phi(x)$ denote an integer polynomial
such that $\phi(n) >0$ for all $n>0$. Consider the map $S_{\phi,b}: {\mathbb
Z}_{\geq 0} \to {\mathbb Z}_{\geq 0}$, with $ S_{\phi,b}(n) := \phi(x_0)+ \dots
+ \phi(x_d)$. It is known that the orbit set $\{n,S_{\phi,b}(n),
S_{\phi,b}(S_{\phi,b}(n)), \dots \}$ 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 $\ell\geq 1$ and let $\phi(x)=x^2$. We show that the set
of bases $b\geq 2$ 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.
