Published January 15, 2025 | Version 1.3
Preprint Restricted

The Collatz Conjecture: A Proof Through Series Interconnections and Structural Properties of Power-of-2 Generated Odd Number Series

Creators

Description

This work introduces and analyzes a mathematical structure based on numerical series generated by multiplying odd numbers by powers of 2. It is demonstrated that these series form a natural partition of positive integers and are interconnected through a generating function f(x) = (x-1)/3.

The study establishes the uniqueness of connection sequences between different series, proves the existence of a unique cycle in the system, and provides a rigorous demonstration of the convergence of every sequence toward the fundamental series S₁. Furthermore, a conservation property is proved regarding the sum of differences between consecutive elements in the sequence.


Finally, we demonstrate how this mathematical structure provides a complete proof of the Collatz conjecture through the bijective properties of series connections and the uniqueness of convergence paths.

Table of contents (English)

 

  1. Introduction
    • 1.1 Motivation
    • 1.2 Research Objectives
  2. Methods
  3. Structure of the Series
    • 3.1 Definitions
    • 3.2 Proposition (Uniqueness of Elements)
    • 3.3 Proposition (Coverage)
    • 3.4 Proposition (Disjoint Series)
  4. Generating Function for Odd Numbers
    • 4.1 Definitions
    • 4.2 Proposition (Bijective Correspondence)
    • 4.3 Theorem (Generators of Odd Numbers in Sd Series)
      • 4.3.1 Lemma (Modulo 3 Pattern in Generated Numbers)
    • 4.4 Theorem (Generation Properties in Sd Series)
    • 4.5 Lemma of Generated Numbers Modulus (Case n = 1)
    • 4.6 Theorem (Modular Patterns in Generated Numbers)
      • 4.6.1 Proposition (Generation Pattern)
      • 4.6.2 Lemma (Properties of Squares in Number Generation)
    • 4.7 Proposition (Different Series)
    • 4.8 Theorem (Uniqueness of Loop in S₁)
      • 4.8.1 Proposition (Coverage and Uniqueness of Sequences)
    • 4.9 Theorem (Convergence to S₁ and Conservation of Differences)
      • 4.9.1 Lemma (Physical Interpretation of Series Transitions)
    • 4.10 Theorem (Sequence Uniqueness)
    • 4.11 Theorem (Convergence through Bijectivity)
    • 4.12 Theorem (Equivalence with Collatz Conjecture)
  5. Main Results
  6. Discussion
  7. Conclusions

Methods (English)

The methodological approach is based on:

  1. Elementary number theory, particularly:
    • Properties of powers of 2
    • Congruences modulo 2 and 3
    • Properties of odd numbers
  2. Proofs by contradiction and bijectivity
  3. Analysis of conservation properties in sequences
  4. Study of generating functions between sets of natural numbers
  5. Analysis of the equivalence between series transitions and the Collatz function through bijective mappings and structural correspondence

Files

Restricted

The record is publicly accessible, but files are restricted to users with access.

Additional details

Related works

Describes
Model: 10.5281/zenodo.14680949 (DOI)

Dates

Created
2024-12-23
Available
2025-01-09
Updated
2025-01-12
Structural Changes to Theorem 4.9 (Convergence and Conservation Properties)
Updated
2025-01-13
Structural Changes to Theorem 4.9 Theorem (Convergence to S1 and Conservation of Differences)
Updated
2025-01-15
Added 4.8.1 Proposition (Coverage and Uniqueness of Sequences)