Abstract
This manuscript focuses on one of the most famous open problems in mathematics, namely the Collatz conjecture. The first part of the paper is devoted to describe the problem, providing a historical introduction to it, as well as giving some intuitive arguments of why is it hard from the mathematical point of view. The second part is dedicated to the visualization of behaviors of the Collatz iteration function and the analysis of the results.
Highlights
The visualization of the Hailstone sequences is often performed by means of directed graphs. While these representations produce simple to read plots, the question arises on how the ‘rules’ adopted for the graphical representation put some additional conditions on the final plot. Having this idea in mind, we propose the adoption of two clustering computational techniques, namely the hierarchical clustering (HC) and multidimensional scaling (MDS) methods, for computational clustering and visualization [13,14,15,16,17,18,19,20,21]
This paper proposes a clustering perspective to analyze the Collatz conjecture
Hailstone sequences were analyzed by means of clustering techniques, namely the HC and MDS computational algorithms
Summary
It is possible to study and provide representations of the second possibility in Remark 1 Observe that, essentially, this second possibility deals with the existence of a positive integer solution to certain linear equations modulo 2 N. This second possibility deals with the existence of a positive integer solution to certain linear equations modulo 2 N This idea is developed, and it is used to describe some interesting relations and graphical representations. The visualization of the Hailstone sequences is often performed by means of directed graphs While these representations produce simple to read plots, the question arises on how the ‘rules’ adopted for the graphical representation put some additional conditions on the final plot.
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