Validating Collatz Conjecture Through Binary Representation and Probabilistic Path Analysis

EasyChair Preprint 12148

Abstract

The Collatz conjecture, a longstanding mathematical puzzle, posits
that, regardless of the starting integer, iteratively applying a specific formula
will eventually lead to the value 1. This paper introduces a novel
approach to validate the Collatz conjecture by leveraging the binary representation
of generated numbers. Each transition in the sequence is predetermined
using the Collatz conjecture formula, yet the path of transitions
is revealed to be intricate, involving alternating increases and decreases
for each initial value.
The study delves into the global flow of the sequence, investigating the
behavior of the generated numbers as they progress toward the termination
value of 1. The analysis utilizes the concept of probability to shed
light on the complex dynamics of the Collatz conjecture. By incorporating
probabilistic methods, this research aims to unravel the underlying
patterns and tendencies that govern the convergence of the sequence.
The findings contribute to a deeper understanding of the Collatz conjecture,
offering insights into the inherent complexities of its trajectories.
This work not only validates the conjecture through binary representation
but also provides a probabilistic framework to elucidate the global flow of
the sequence, enriching our comprehension of this enduring mathematical
mystery.

Keyphrases: Binary Number, Collatz Conjecture, probabilistic methods

@booklet{EasyChair:12148,
  author    = {Budee U Zaman},
  title     = {Validating Collatz Conjecture Through Binary Representation and Probabilistic Path Analysis},
  howpublished = {EasyChair Preprint 12148},
  year      = {EasyChair, 2024}}