2025/06/29 by Youchun Luo, Luo, Youchun
Computer Science · Mathematics · #Analytic Number Theory Research #Benford’s Law and Fraud Detection #math.GM #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2506.23070
openalex publication_date 2025/06/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
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., \frac2E( 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 ) =\lfloor log 62⋅ D( N ) -log 6N \rfloor (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.