On the Glide of 3x+1 Problem

TL;DR

This paper investigates the 3x+1 problem, establishing inequalities related to the iterated function, and confirms a conjecture by Terras from 1976 regarding the behavior of the sequence.

Contribution

It proves a conjecture by Terras, relating the number of odd and even steps in the 3x+1 sequence, providing new bounds and insights into the problem's dynamics.

Findings

Confirmed Terras's conjecture from 1976.

Established inequalities linking odd/even steps to sequence behavior.

Provided bounds on the iterated function's growth.

Abstract

For any positive integer $n$, define an iterated function $$ f(n)=\left\{\begin{array}{ll} n/2, & \mbox{$n$ even,} \\ 3n+1, & \mbox{$n$ odd.} \end{array} \right. $$ Suppose $k$ (if it exists) is the lowest number such that $f^{k}(n)<n$, and there are $O(n)$ "multiply by three and add one" and $E(n)$ "divide by two" from $n$ to $f^{k}(n)$, then there must be $$ 2^{E(n)-1}<3^{O(n)}<2^{E(n)}. $$ Our results confirm the conjecture proposed by Terras in 1976.