Fixed Points and Cycles of the Kaprekar Transformation: 2. Even bases

TL;DR

This paper classifies fixed points and cycles of the Kaprekar transformation in even bases, revealing their structure through group theory and providing enumeration methods, with detailed analysis for bases 4, 6, and 8.

Contribution

It introduces a classification of fixed points and cycles for even bases and offers formulas for enumeration, extending understanding of Kaprekar's transformation.

Findings

Complete classification in base 4.

Cycle structures determined by subgroup cosets.

Enumeration methods for fixed points and cycles.

Abstract

We develop a classification of the fixed points and cycles of the Kaprekar transformation in even bases. The most numerous fixed points and cycles are those we denote symmetric and almost-symmetric; the structure of the cycles of these classes in base $b$ is determined by subgroups and cosets in the multiplicative group modulo $b-1$. We provide methods and formulae for enumerating the fixed points and cycles of these and other classes. A detailed survey of the fixed points and cycles is provided for bases 4, 6 and 8, including a rigorous proof that the classification is complete in base 4.