# Convergence of generalized Collatz problem in k-adic field

## TL;DR

This paper introduces a new k-adic series transformation called Z-transformation, explores its fixed points and periodicity, and extends the period problem to broader k-adic fields, revealing specific periodic sequences.

## Contribution

It defines the Z-transformation, analyzes its periodicity in k-adic fields, and extends the Collatz problem to more complex algebraic systems.

## Key findings

- Periodic sequences M1={1,2} and M2={1,2}∪{n_0}∪{n'} identified
- Different constraints on k produce distinct periodic columns
- Extension of the Collatz problem to broader algebraic systems

## Abstract

In this article, we define a new k-adic series transformation called Z-transformation and probe into its fixed point and periodicity. We extend the number field of the transform period problem to a wider k-adic field. Different constraints are imposed on k, then different periodic columns are formed after finite Z-transformations. We obtain that their periodic sequences are M1= {1,2} and M2={1,2}\cup {n_0}\cup {n'} respectively after derivation. As an application, it can provide a reference for C problems in more complex algebraic ystems.