On the orbits associated with the Collatz conjecture

TL;DR

This paper investigates the properties of Collatz matrices and their determinants, aiming to understand the orbit structure related to the Collatz conjecture, and improves upon previous partial results in this mathematical problem.

Contribution

The article advances the analysis of Collatz matrices by extending the calculation of determinants and addressing cases previously unresolved, contributing to the orbit-based approach to the Collatz conjecture.

Findings

Calculated determinants for initial Collatz matrices.

Identified limitations in proving determinant invariance for certain matrix dimensions.

Improved understanding of the orbit structure related to the Collatz conjecture.

Abstract

This article is based upon previous work by Sousa Ramos and his collaborators. They first prove that the existence of only one orbit associated with the Collatz conjecture is equivalent to the determinant of each matrix of a certain sequence of matrices to have the same value. These matrices are called Collatz matrices. The second step in their work would be to calculate this determinant for each of the Collatz matrices. Having calculated this determinant for the first few terms of the sequence of matrices, their plan was to prove the determinant of the current term equals the determinant of the previous one. Unfortunately, they could not prove it for the cases where the dimensions of the matrices are 26+54l or 44+54l, where l is a positive integer. In the current article we improve on these results.