Collatz Iterative Trajectories of All Odd Numbers Attain Bounded Values  

作　　者：Jinqing Zhang Xintong Zhang Jinqing Zhang;Xintong Zhang(Beijing, China;Toronto, Canada)

机构地区：[1]Beijing, China [2]Toronto, Canada

出　　处：《Journal of Applied Mathematics and Physics》2023年第10期3030-3041,共12页应用数学与应用物理（英文）

摘　　要：The aim of this paper is to study the 3x + 1 problem based on the Collatz iterative formula. It can be seen from the iterative formula that the necessary condition for the Collatz iteration convergence is that its slope being less than 1. An odd number N that satisfies the condition of a slope less than 1 after n^th Collatz iterations is defined as an n-step odd number. Through statistical analysis, it is found that after n^th Collatz iterations, the iterative value of any n-step odd number N that is greater than 1 is less than N, which proves that the slope less than 1 is a sufficient and necessary condition for Collatz iteration convergence.The aim of this paper is to study the 3x + 1 problem based on the Collatz iterative formula. It can be seen from the iterative formula that the necessary condition for the Collatz iteration convergence is that its slope being less than 1. An odd number N that satisfies the condition of a slope less than 1 after n^th Collatz iterations is defined as an n-step odd number. Through statistical analysis, it is found that after n^th Collatz iterations, the iterative value of any n-step odd number N that is greater than 1 is less than N, which proves that the slope less than 1 is a sufficient and necessary condition for Collatz iteration convergence.

关 键 词：Number Theory 3x + 1 Problem Collatz Conjecture Syracuse Problem Statistical Analysis 

分 类 号：O17[理学—数学]