Iteration Steps of 3x+1 Problem

Youchun Luo

mathematics

Iteration Steps of 3x+1 Problem

arXiv:2506.23070

summary

The paper analyzes the numbers of total, odd, and even iteration steps in the 3x+1 (Collatz) problem, proposes a Weak Residue Conjecture, and derives explicit formulas linking these step counts under the conjectures, also extending the analysis to a qx+1 variant.

Abstract

On the 3x+1 problem, given a positive integer $N$ , let $D (N)$ , $O (N)$ and $E (N)$ denote the total number of iteration steps, the number of odd iteration steps, and the number of even iteration steps, respectively, when $N$ is iterated until it reaches 1. It is straightforward to observe that $D (N) = O (N) + E (N)$ . In this paper, we propose a conjecture termed the Weak Residue Conjecture(i.e., $\frac{2 ^{E (N)}}{3 ^{O (N)} \cdot N} < 2$ ). We prove that if the 3x+1 conjecture is true and the Weak Residue Conjecture is true, there exist nontrivial relationships among $D (N)$ , $O (N)$ , $E (N)$ , i.e., $O (N) = ⌊ lo g_{6} 2 \cdot D (N) - lo g_{6} N ⌋$ (this implies that, given $N$ , both $O (N)$ and $E (N)$ can be directly computed from $D (N)$ ), and five more similar equations are derived simultaneously. Similarly, the case of qx+1 problem is studied too.

qx+1 situation added

Topics & keywords

#collatz conjecture #iteration steps #odd-even steps #weak residue conjecture #integer dynamics3x+1 problemtotal steps D(N)odd steps O(N)even steps E(N)log_6 relationshipqx+1 variant