Collatz mapping on $\mathbb{Z}/10\mathbb{Z}$

TL;DR

This paper explores the Collatz conjecture within the modular arithmetic setting of Z/10Z, analyzing its properties and graph structure to gain insights into its behavior.

Contribution

It introduces the Collatz conjecture, develops lemmas to understand its properties modulo 10, and constructs the associated Collatz graph for Z/10Z.

Findings

Properties of the Collatz graph on Z/10Z are characterized.

Lemmas supporting the structure of the Collatz conjecture are established.

The graph visualization provides new perspectives on the conjecture's behavior.

Abstract

We introduce the \emph{Collatz conjecture} and its history. Some definition that this conjecture has, will be expressed and with these we try to explain some good lemma to justify the main properties of the \emph{Collatz conjecture}. With these lemmas a situation made to go through to write some mainly properties of Collatz graph on $\mathbb{Z}/10\mathbb{Z}$. Then with something provided it will go to draw the graph on $\mathbb{Z}/10\mathbb{Z}$.