Method Study on the 3x+1 Problem

TL;DR

This paper proposes a probabilistic approach to the 3x+1 problem, suggesting that by analyzing the stopping times and their distribution, the conjecture can be effectively addressed using computational methods.

Contribution

It introduces a novel method based on probabilistic analysis of stopping times, offering a new perspective for solving the 3x+1 conjecture with computational support.

Findings

The approach indicates the decreasing likelihood of counterexamples as calculations increase.

It provides a framework for using large-scale computations to verify the conjecture.

The method suggests the conjecture could be proven by demonstrating the diminishing size of problematic cases.

Abstract

The 3x+1 problem is one of the most classical problems in computer science, related to many fields. As it is thought by scientists a highly hard problem, resolving it successfully not only can improve the research in many relating fields, but also be meaningful to the method study. By deep analyzing the 3x+1 calculation process with the input positive integer becoming greater, we find a useful way for solving this problem with high probability. By making use of the greater calculating ability of great computers and the internet, our way is a valid and powerful way for utterly solving the 3x+1 problem. This way can be expressed in three points: 1) If we can find a positive integer N, for any positive integer less than 2N, the times of dividing 2 out of its stopping time is less than or equal to N, then the 3x+1 conjecture is true; 2) This N may be big, so the calculation may be too big. Our way for solving this is: to find a positive integer K, for all positive integers less than 2K, not all the times of dividing 2 out of their stopping time for these integers are less than or equal to K, some part of these are greater than K, but the number of this part becomes less and less with the K increasing; 3) This K and the calculation may also be too big, our way for solving this is: to find a positive integer R, for all positive integers less than 2R, as above, out of their stopping time, the times of dividing 2 of some part of these integers are greater than R, also the number of this part integers does not become less immediately with the R increasing, but the increasing rate of this number is less and less until to below zero.