Some elements of a possible demonstration of the Collatz conjecture

TL;DR

This paper explores probabilistic properties of the Collatz sequence, demonstrating equal probabilities of increase and decrease at each step, and argues that the conjecture is false for certain linear problems, highlighting the importance of probabilistic analysis.

Contribution

It presents a probabilistic analysis of the Collatz sequence, showing equal increase and decrease probabilities, and claims the conjecture is false for specific linear functions, introducing new perspectives.

Findings

Equal probabilities of increase and decrease in Collatz steps

The Collatz conjecture is false for certain linear functions with parameters a≥5

Probabilistic behavior influences the sequence's tendency to decrease

Abstract

In this paper we are shown the following facts: The probability of increased $ A_{k}=P(T^{k} (x_{0})>T^{k-1} (x_{0})) $, and the probability of decrease $B_{k}=P(T^{k} (x_{0})<T^{k-1} (x_{0}))$ in step $ k $ of a Collataz procedure initiated in $ x_ {0} \in \mathbb{N} $ arbitrary, they are equal for all values of $ k $. This influences on the law that generates the numbers of a Collatz sequence so that it is forced to decrease until the unit. Also is shown in the Collatz conjecture is false for every problem $an +b$ such that $a\geq 5\geq b+2$, and your probabilistic caracter can not be ignored if you want to get to the definitive solution, among other interesting arguments.