Iteration Steps of 3x+1 Problem

Abstract

On the 3x+1 problem, given a positive integer $N$, let $D\left( N \right) $, $O\left( N \right) $ and $E\left( N \right) $ 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\left( N \right) =O\left( N \right) +E\left( N \right) $. In this paper, we propose a conjecture termed the Weak Residue Conjecture(i.e., $\frac{2^{E\left( N \right)}}{3^{O\left( N \right)}\cdot N}<2$). We prove that if the 3x+1 conjecture is true and the Weak Residue Conjecture is true, there exist nontrivial relationships among $D\left( N \right) $, $O\left( N \right) $, $E\left( N \right) $, i.e., $O\left( N \right) =\lfloor \log _62\cdot D\left( N \right) -\log _6N \rfloor $(this implies that, given $N$, both $O\left( N \right) $ and $E\left( N \right) $ can be directly computed from $D\left( N \right) $), and five more similar equations are derived simultaneously. Similarly, the case of qx+1 problem is studied too.