Iteration Steps of 3x + 1 Problem

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., 2E(N)3O(N)·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)=⌊ log 62·D(N)− log 6N⌋ (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.