The 3x+1 Problem and Integer Representations

TL;DR

This paper investigates the 3x+1 problem using set-theoretic and Diophantine methods to represent inverse iterates, revealing that sequences with the trivial cycle can have an arbitrarily small ratio of odd terms.

Contribution

It introduces a set-theoretic approach to represent inverse iterates and provides new insights into the sequence behavior and the ones-ratio in the 3x+1 problem.

Findings

Inverse iterates are represented by exponential Diophantine equations.

Sequences with the trivial cycle can have arbitrarily many odd terms.

The ones-ratio approaches zero in such sequences.

Abstract

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}.