The fundamental properties characterizing the structural behaviors of Collatz sequences

TL;DR

This paper investigates the structural properties of Collatz sequences using binary matrix representations, revealing fundamental behaviors like periodicity and divergence through a novel combinatorial approach.

Contribution

It introduces a new matrix-based framework to analyze Collatz sequences, identifying key properties such as structural periodicity and divergence.

Findings

Identifies structural periodicity in Collatz sequences.

Defines a property of structural divergence.

Provides a binary matrix representation of Collatz behaviors.

Abstract

This work represents an in-depth study of the structural behavior of the Collatz sequences. We consider a finite arithmetic progression with a common difference is 2 and the number of terms in the sequence is equal to 2^n . After, we consider a 2^n x(n+1) matrix ((n+1) columns and 2^n rows) such as the first column contains the terms of arithmetic progression and each row of the matrix represent a finite Collatz sequence. Then, each element of the matrix will be replaced by 0 or 1 according to the following rule: the even integer is replaced by 0 and the odd integer is replaced by 1. We obtained a table contains all binary permutation with repetition this property is called the property of structural complementarily. Based on these tables, we can determine any other fundamentals properties characterizing the behavior of Collatz sequences. Thus, we can distinguish two other important properties such as the property of structural cyclical behavior characterized by a structural periodicity and the structural sliding property that can be called also the property of structural divergence.