The Collatz Network

TL;DR

This paper constructs a graph representing all possible Collatz paths to analyze the conjecture, introduces an extended version with divergent orbits, and explores connections between Fibonacci numbers and primes.

Contribution

It introduces a graph-based framework for Collatz paths, defines an extended conjecture with divergent orbits, and links Fibonacci sequences to prime numbers.

Findings

The minimal element of Collatz orbits is 1.

Infinite divergent orbits exist in the extended Collatz conjecture.

Theorems relate Fibonacci numbers to prime numbers.

Abstract

Considering all possible paths that a natural number can take following the rules of the algorithm proposed in the Collatz conjecture we construct a graph that can be interpreted as an infinite network that contemplates all possible paths within the conjecture. This allows us to understand why the minimal element of the Collatz orbit $x$ is equal to 1. Subsequently we define the extended Collatz conjecture equal to $o x+1$ when $x$ is odd and $x/2$ when $x$ is even with $o$ an odd number greater than $3$ and we show that there are infinite orbits in the extended Collatz conjecture that diverges. Finally, we find interesting theorems relating the Fibonacci sequence to prime numbers.