The Analysis of Convergence for the 3<i>X</i>+ 1 Problem and Crandall Conjecture for the a<i>X</i>+ 1 Problem

作     者：Zuojun Hu Zuojun Hu

作者机构：Huadian Jinsha River Upstream Hydropower Development Co. Ltd. Chengdu China 

出 版 物：《Advances in Pure Mathematics》 (理论数学进展（英文）)

年 卷 期：2021年第11卷第5期

页      面：400-407页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

主　　题：3X + 1 Problem Crandall Conjecture Energy Method Expectancy Theory 

摘      要：The 3X + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of the 3X + 1 problem. It is worth noting that, both conjectures are infamous for their simplicity in stating but intractability in solving. In this paper, I aim to provide a clear explanation about the reason why these two problems are difficult to handle and have very different characteristics on convergence of the series via creatively applying the probability theory and global expectancy value E(n) of energy contraction index. The corresponding convergence analysis explicitly shows that a = 3 leads to a difficult problem, while a 3 leads to a divergent series. To the best of my knowledge, this is the first work to point out the difference between these cases. The corresponding results not only propose a new angle to analyze the 3X + 1 problem, but also shed some light on the future research.