Abstract
Let [Formula: see text] be an integer, and write the base [Formula: see text] expansion of any non-negative integer [Formula: see text] as [Formula: see text], with [Formula: see text] and [Formula: see text] for [Formula: see text]. Let [Formula: see text] denote an integer polynomial such that [Formula: see text] for all [Formula: see text]. Consider the map [Formula: see text], with [Formula: see text]. It is known that the orbit set [Formula: see text] is finite for all [Formula: see text]. Each orbit contains a finite cycle, and for a given [Formula: see text], the union of such cycles over all orbit sets is finite. Fix now an integer [Formula: see text] and let [Formula: see text]. We show that the set of bases [Formula: see text] which have at least one cycle of length [Formula: see text] 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.
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