The Prime-Power Map

TL;DR

This paper studies a modified prime map called the prime-power map, analyzing its iterative behavior over positive integers, comparing it to a simpler variant, and conjecturing about the nature of its orbits based on experimental data.

Contribution

It introduces and analyzes the prime-power map, providing new results on its dynamics and comparing it to a simplified version, with conjectures supported by experimental observations.

Findings

Parallel results to the prime map are established.

A comparison with a manageable variant reveals differences in orbit structure.

Conjecture that almost all orbits contain no prime-power.

Abstract

We introduce a modification of Pillai's prime map: the prime-power map. This map fixes $1$, divides its argument by $p$ if it is a prime-power $p^k$, otherwise subtracts from its argument the largest prime-power not exceeding it. We study the iteration of this map over the positive integers, developing, firstly, results parallel to those known for the prime map. Subsequently, we compare its dynamical properties to those of a more manageable variant of the map under which any orbit admits an explicit description. Finally, we present some experimental observations, based on which we conjecture that almost every orbit of the prime-power map contains no prime-power.