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International Journal of Pure Mathematical Sciences
IJPMS Volume 21
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Divisions by Two in Collatz Sequences: A Data Science Approach

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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.

Info:

Periodical:
International Journal of Pure Mathematical Sciences (Volume 21)
Pages:
1-13
Citation:
C. Koch et al., "Divisions by Two in Collatz Sequences: A Data Science Approach", International Journal of Pure Mathematical Sciences, Vol. 21, pp. 1-13, 2020
Online since:
November 2020
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References:

[1] J. C. Lagarias: The Ultimate Challenge: The 3x+1 Problem. American Mathematical Society, 2010, ISBN 978­0821849408.

[2] E. Sultanow, C. Koch and S. Cox: Collatz Sequences in the Light of Graph Theory (Fourth Ver­ sion). University of Potsdam, 2020, DOI https://doi.org/10.25932/publishup­44325.

[3] R. Sedgewick and K. Wayne: Algorithms (Fourth Edition). Addison­Wesley Professional, 2011, ISBN 978­0321573513.

[4] T.M.M. Laarhoven: The 3n + 1 conjecture. Eindhoven University of Technology, July (2009).

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