Parametric transformation functions in the Kaprekar routine I

TL;DR

This paper introduces parametric functions to analyze the Kaprekar routine, enabling the study of its algebraic structure, constants, and cycles through a novel parametric approach.

Contribution

It develops Ki functions for the Kaprekar transformation, facilitating algebraic analysis and cycle detection in the iterative process.

Findings

Derived constants and cycles using the parametric functions

Analyzed the algebraic structure of transformation trees

Provided a framework for further theoretical exploration

Abstract

The Kaprekar transformation is uniquely determined by parameters based on differences between symmetric values in the ordered numeric sequence. The parametric analysis of the iteration of the process requires developing Ki functions that provide the parameters of the transformed number. This is the objective of this work. As application examples we use these functions to deduce constants and cycles. But its main utility is to study the algebraic architecture of the transformation trees that we develop in the second part of this work.