The Collatz Conjecture “The Non-Existence of Divergent Orbits Through an Analysis of the Codification of Linear Diophantine Dynamical Systems”

We define the function $Col: \mathbb{N} \to \mathbb{N}$ as the Collatz function, given by $3n + 1$ if $n$ is odd and $\displaystyle\frac{n}{2}$ if $n$ is even. The conjecture postulates that for any positive integer, at some point, its iteration will reach 1, or equivalently, every orbit will fall into the periodic cycle $\{4, 2, 1\}$. Two conditions would invalidate the conjecture: The existence of a divergent orbit or the presence of another cycle. We can study the dynamics of the orbits through the density of even terms in their orbit. If all points' accumulation density exceeds the value of $\displaystyle\frac{\ln(3)}{\ln(2)}$ then the orbit is bounded. The main result of this work is to show that there are no natural numbers such that the accumulation points of the pair density are less than $\displaystyle\frac{\ln(3)}{\ln(2)}$. In other words, there are no divergent orbits.