
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 is equal to 1. Subsequently we define the extended Collatz conjecture equal to when is odd and when is even with an odd number greater than 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.
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Taxonomy
TopicsBenford’s Law and Fraud Detection · Digital Media Forensic Detection
