Iteration Steps of 3x+1 Problem

TL;DR

This paper explores the iteration steps of the 3x+1 problem, proposing a weak residue conjecture and deriving relationships among iteration counts assuming the conjecture and the 3x+1 conjecture.

Contribution

It introduces the weak residue conjecture and establishes new formulas linking total, odd, and even iteration steps under certain assumptions.

Findings

Proposes the weak residue conjecture for the 3x+1 problem.

Derives formulas relating iteration counts assuming conjectures hold.

Shows how to compute odd and even steps from total steps using derived equations.

Abstract

On the 3x+1 problem, given a positive integer $N$, let $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $ be the total iteration steps, the odd iteration steps and the even iteration steps when $N$ iterates to 1(except 1) respectively. Trivially, we have $D\left( N \right) =O\left( N \right) +E\left( N \right) $. In this paper, we propose a so-called weak residue conjecture(i.e., $\frac{2^{E\left( N \right)}}{3^{O\left( N \right)}\cdot N}\le 2$). We prove that if 3x+1 conjecture is true and the weak residue conjecture is true, there exist non-trivial relationships among $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $, i.e., $O\left( N \right) =\lfloor \log_6 2\cdot D\left( N \right) -\log_6 N \rfloor $(it implies that we can calculate $O\left( N \right) $, $E\left( N \right) $ directly by $D\left( N \right) $ only, of course given $N$), and 5 more similar equations are derived simultaneously.