The 3x+1 Problem and Integer Representations

The $3x+1$ Problem asks if whether for every natural number $n$, there exists a finite number of iterations of the piecewise function $$f(2n)=n, \quad f(2n-1)=6n-2,$$ with an iterate equal to the number $1$, or in other words, every sequence contains the trivial cycle $\left\langle {4,2,1}\right\rangle$. We use a set-theoretic approach to get representations of all inverse iterates of the number $1$. The representations, which are exponential Diophantine equations, help us study both the \textit{mixing} property of $f$ and the asymptotic behavior of sequences containing the trivial cycle. Another one of our original results is the new insight that the \textit{ones-ratio} approaches zero for such sequences, where the number of odd terms is \textit{arbitrarily large}.Comment: 28 page

Integer Representations and Trajectories of the 3x+1 Problem

This paper studies certain trajectories of the Collatz function. I show that if for each odd number $n$, $n\sim 3n+2$ then every positive integer $n \in \mathbb{N}\setminus 2^{\mathbb{N}}$ has the representation $$n=\left(2^{a_{k+1}}-\sum_{i=0}^{k}{2^{a_i}3^{k-i}}\right)/ 3^{k+1}$$ where $0\le a_0 \le a_1 \le \cdot \cdot \cdot \le a_{k+1}$. As a consequence, in order to prove Collatz Conjecture I illustrate that it is sufficient to prove $n\sim 3n+2$ for any odd $n\in \mathbb{N}\setminus 2^{\mathbb{N}}$. This is the main result of the paper

An introduction to pp-adic systems: A new kind of number systems inspired by the Collatz 3n+13n+1 conjecture

This article introduces a new kind of number systems on $p$-adic integers which is inspired by the well-known $3n+1$ conjecture of Lothar Collatz. A $p$-adic system is a piecewise function on $\mathbb{Z}_p$ which has branches for all residue classes modulo $p$ and whose dynamics can be used to define digit expansions of $p$-adic integers which respect congruency modulo powers of $p$ and admit a distinctive "block structure". $p$-adic systems generalize several notions related to $p$-adic integers such as permutation polynomials and put them under a common framework, allowing for results and techniques formulated in one setting to be transferred to another. The general framework established by $p$-adic systems also provides more natural versions of the original Collatz conjecture and first results could be achieved in the context. A detailed formal introduction to $p$-adic systems and their different interpretations is given. Several classes of $p$-adic systems defined by different types of functions such as polynomial functions or rational functions are characterized and a group structure on the set of all $p$-adic systems is established, which altogether provides a variety of concrete examples of $p$-adic systems. Furthermore, $p$-adic systems are used to generalize Hensel's Lemma on polynomials to general functions on $\mathbb{Z}_p$, analyze the original Collatz conjecture in the context of other "linear-polynomial $p$-adic systems", and to study the relation between "polynomial $p$-adic systems" and permutation polynomials with the aid of "trees of cycles" which encode the cycle structure of certain permutations of $\mathbb{Z}_p$. To outline a potential roadmap for future investigations of $p$-adic systems in many different directions, several open questions and problems in relation to $p$-adic systems are listed.Comment: Added missing results of computations on pages 60 and 6