Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The theory of the Collatz process and the method of dynamical balls

Theophilus Agama

Abstract

In this paper, we introduce and develop the theory of the Collatz process and the method of dynamical balls. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie Germain primes. We provide several formulations of the Collatz conjecture in this language. Furthermore, we introduce and develop the notion of dynamical systems induced by fixed $a\in \mathbb{N}$ and their associated induced dynamical balls. We develop tools to study problems that require determining the convergence of certain sequences generated by iterating on a fixed integer.

The theory of the Collatz process and the method of dynamical balls

Abstract

In this paper, we introduce and develop the theory of the Collatz process and the method of dynamical balls. We leverage this theory to study the Collatz conjecture. This theory also has a subtle connection with the infamous problem of the distribution of Sophie Germain primes. We provide several formulations of the Collatz conjecture in this language. Furthermore, we introduce and develop the notion of dynamical systems induced by fixed and their associated induced dynamical balls. We develop tools to study problems that require determining the convergence of certain sequences generated by iterating on a fixed integer.

Paper Structure

This paper contains 13 sections, 24 theorems, 91 equations.

Table of Contents

  1. Introduction and motivation
  2. Organization of the paper
  3. Modified Collatz function and the Collatz process
  4. The order and index under the Collatz process
  5. Relative speed of the Collatz process
  6. The sub-Collatz process
  7. Distribution of the Collatz process
  8. Dynamical systems induced by sequences
  9. Analysis on dynamical balls
  10. The limit of dynamical balls
  11. Dynamical waves and amplitude of waves induced by dynamical balls
  12. Dynamical waves estimate
  13. Translation and dilation of dynamical balls

Key Result

Proposition 2.3

Let $f$ be the Collatz function with the corresponding Collatz process $\{f^{s}(b)\}_{s=1}^{\infty}$. If $b$ is the generator, then each $a_{n}\in \{\mathrm{Inf}\{f^{-s}(b)\}\}_{s=1}^{\infty}$ must satisfy

Theorems & Definitions (77)

  • Definition 1.1
  • Conjecture 1.2
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • ...and 67 more