The Collatz problem (a*q+-1,a=1,3,5,...) from the point of view of transformations of Jacobsthal numbers

TL;DR

This paper investigates the Collatz problem using Jacobsthal number transformations, revealing new branching rules and trajectory behaviors, especially for functions with factors 3 and 5, and proposes an analytical model linking iteration steps.

Contribution

It introduces a novel approach by analyzing Collatz trajectories through Jacobsthal number transformations and formulates an analytical model for their development.

Findings

Collatz trajectories form in the reverse Jacobsthal tree direction.

Sequences with factors 3 and 5 do not always reach the unit element.

An analytical model relates iteration steps to Jacobsthal number properties.

Abstract

In the paper, from the point of view of recurrent numbers of the Jacobsthal type, the Collatz problem with the general aq+-1 function of conjecture odd positive integers q from the set of natural numbers is investigated. Formulated branching rules from nodes with generalized Jacobsthal numbers of the so-called Jacobsthal tree. It is shown that Collatz trajectories are formed in the reverse direction of the Jacobsthal tree. It is shown that unlike the classical Collatz problem, in which the Collatz sequence for a finite number of iterations leads to the unit element, for functions with factors 3 and 5, the Collatz sequence for a finite number of iterations does not reach the unit element. An analytical model is proposed that relates the iteration number of doubling (or halving the number in the reverse direction) number between two odd-numbered branch nodes. It is shown that for an arbitrary number, Collatz and Jacobsthal trajectories develop in the respective directions, and the Jacobsthal trajectory for a finite number of iterations leads to an element that is a multiple of the factor a in the function aq+-1.