Characteristic numbers and characteristic equations of parity vectors of Collatz sequences

TL;DR

This paper introduces a novel framework using characteristic numbers and equations to analyze the parity vectors of Collatz sequences, aiding in understanding their behavior and divergence properties.

Contribution

It develops the concept of characteristic numbers and equations for parity vectors, linking finite and infinite sequences, and explores conditions for sequence divergence.

Findings

Characterizes parity vectors using characteristic numbers.

Establishes relations between sequence start terms and parity vectors.

Identifies conditions for the existence or non-existence of divergent sequences.

Abstract

The present work deals with the characterization of parity vectors of Collatz sequences (of finite and infinite length). Such a characterization leads to the determination of several numbers (integers or non-integers) that we call the characteristic numbers of a given parity vector. Some characteristic numbers are linked together by equations that can be called characteristic equations of the considered parity vector. If a parity vector v of finite length n contains the first n terms of a parity vector V of infinite length then all the characteristic numbers of v are considered as characteristic numbers of order n of the infinite vector V. The limits of nth order characteristic numbers when n tends to infinity constitute the absolute characteristic numbers of the infinite vector V and they allow determining its behavior and properties. In this paper, we study the properties of parity vectors of infinite length based on their characteristic numbers. Then, we establish the relations between the first term of a Collatz sequence and its parity vector. Finally, still based on the characteristic numbers, we determine some conditions of existence and non-existence of divergent sequences.