Divisions by Two in Collatz Sequences: A Data Science Approach

Christian Koch,Sean Cox,Eldar Sultanow

doi:10.18052/www.scipress.com/ijpms.21.1

Abstract

The Collatz conjecture is an unsolved number theory problem. We approach the question by examining the divisions by two that are performed within Collatz sequences. Aside from classical mathematical methods, we use techniques of data science. Based on the analysis of 10,000 sequences we show that the number of divisions by two lies within clear boundaries. Building on the results, we develop and prove an equation to calculate the maximum possible number of divisions by two for any given a Collatz sequence. Whenever this maximum is reached, a sequence leads to the result one, as conjectured by Lothar Collatz. Furthermore, we show how many divisions by two are required for a cycle of a specific length. The findings are valuable for further investigations and could form the basis for a comprehensive proof of the conjecture.

Highlights

The ProblemThe Collatz conjecture is a wellknown number theory problem and is the subject of numerous pub lications.[1]
We approach the question by examining the divisions by two that are performed within Collatz sequences
Theorem 16 determines the number of divisions by two that can lead to a cycle

Summary

Introduction

The Collatz conjecture is a wellknown number theory problem and is the subject of numerous pub lications.[1]. Koch and Cox demonstrated that odd numbers of Collatz sequences can be calculated with the following recursive equation:[3] vn+1. Being compliant with the Collatz conjecture, the algorithms 2 and 3 halt if at least one of the following conditions is fulfilled: 1. Applying equation 3 yields: In the above example the algorithm halts after two iterations because the first condition is fulfilled. We know that in every iteration of the equations 2 and 3 at least one division by two is performed This follows from the constraints of the Collatz problem. For a better understanding of the above term, let us consider the example k = 3 and v1 = 5 In this case equation 5 results in: α1 = α1 = 4 = log2 3 + log2 5 + log[2]. The number of divisions by two can be determined with the following formula and will be symbolised by α:

Source

Summary