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Tao, T. (2022) Almost All Orbits of the Collatz Map Attain Almost Bounded Values. Forum of Mathematics, Pi, 10, e12.
https://doi.org/10.1017/fmp.2022.8

has been cited by the following article:

• TITLE: Collatz Iterative Trajectories of All Odd Numbers Attain Bounded Values

  AUTHORS: Jinqing Zhang, Xintong Zhang

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

  JOURNAL NAME: Journal of Applied Mathematics and Physics, Vol.11 No.10, October 27, 2023

  ABSTRACT: 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 nth Collatz iterations is defined as an n-step odd number. Through statistical analysis, it is found that after nth 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.