Author(s)

Jinqing Zhang¹, Xintong Zhang²

¹Beijing, China.
²Toronto, Canada.

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.

Number Theory, 3x + 1 Problem, Collatz Conjecture, Syracuse Problem, Statistical Analysis

Zhang, J.Q. and Zhang, X.T. (2023) Collatz Iterative Trajectories of All Odd Numbers Attain Bounded Values. Journal of Applied Mathematics and Physics, 11, 3030-3041. doi: 10.4236/jamp.2023.1110200.