Exploring the Collatz Conjecture Through Directed Graphs

EasyChair Preprint 12334

Abstract

The Collatz conjecture is a well-known number theory puzzle that
states that every positive integer would eventually converge to the trivial
cycle of 1, 2, 1, 2,... when repeatedly exposed to a particular transformation.
In this transformation, even numbers are divided in half, odd
numbers are tripled, and one is added. In this study, we present a new
method for creating a directed graph and using it to display and analyze
Collatz sequences. Our technique creates what we call a Collatz directed
graph by joining an endless number of simple directed graphs, each of
which corresponds to a natural number. We show by careful mathematical
analysis that all positive integers are included in this Collatz directed
graph. Moreover, we give an evidence that verifies the Collatz conjecture
by showing that the sole cycle in this graph is the trivial cycle of 1, 2, 1,
2,... We also prove that there is no sequence that diverges to infinity in
this graph. Our results provide insights into the fundamental structure of
Collatz sequences and further our knowledge of the Collatz conjecture .

Keyphrases: Collatz Conjecture, directed graphs, intgers

@booklet{EasyChair:12334,
  author    = {Budee U Zaman},
  title     = {Exploring the Collatz Conjecture Through Directed Graphs},
  howpublished = {EasyChair Preprint 12334},
  year      = {EasyChair, 2024}}