# The Periodicity to a Kind of Generalized Collatz Problem

## TL;DR

This paper extends the Collatz problem to k-adic systems, introducing the Z transformation sequence and proving its limit set is {1,2} under certain conditions, contributing to understanding generalized Collatz dynamics.

## Contribution

It defines a new Z transformation sequence for k-adic Collatz generalizations and proves its limit set is {1,2} under specific assumptions, advancing the theoretical understanding.

## Key findings

- Limit set of Z transformation sequence is {1,2}
- Extension of Collatz problem to k-adic systems
- Provides theoretical proof under certain assumptions

## Abstract

The Collatz problem is related to the fixed point problem, and is widely used in mathematics. It has attracted a wide range of math enthusiasts, but is still difficult to solve. So, this article aimed to study the extension of the Collatz problem, more widely, in k-adic. We define a new sequence called Z transformation sequence. Under a suitable assumptions, we can prove that the limit set of the Z transformation sequence must be M={1,2}.