Integer Dynamics

Abstract

By repeatedly summing the squares of the digits in base $b$, we obtain a sequence of integers. In this paper, we are concerned with the cycles that arise in this iterative process. It is known that any such sequence ends in a cycle, and for a fixed base $b$, there are only finitely many cycles. We show that for any $\ell \ge 1$, the set of bases $b$ that admit a cycle of length $\ell$ contains an arithmetic progression, and therefore has positive lower density.