Some elements of a possible demonstration of the Collatz conjecture

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 $a\ge 5\ge b+2$, and your probabilistic caracter can not be ignored if you want to get to the definitive solution, among other interesting arguments.