Article
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Collatz Conjecture Is Analogous to an Inverse Function of Natural Number
Version 1
: Received: 5 May 2024 / Approved: 6 May 2024 / Online: 7 May 2024 (03:03:05 CEST)
Version 2 : Received: 11 May 2024 / Approved: 4 June 2024 / Online: 7 June 2024 (11:04:40 CEST)
Version 2 : Received: 11 May 2024 / Approved: 4 June 2024 / Online: 7 June 2024 (11:04:40 CEST)
How to cite: Feng, J. Collatz Conjecture Is Analogous to an Inverse Function of Natural Number. Preprints 2024, 2024050310. https://doi.org/10.20944/preprints202405.0310.v2 Feng, J. Collatz Conjecture Is Analogous to an Inverse Function of Natural Number. Preprints 2024, 2024050310. https://doi.org/10.20944/preprints202405.0310.v2
Abstract
We introduce a full binary directed tree structure to represent the set of natural numbers, further categorizing them into three distinct subsets: pure odd numbers, pure even numbers, and mixed numbers. We adopt a binary string representation for natural numbers and elaborate on the composite methodology encompassing odd- and even-number functions. Our analysis focuses on examining the iteration sequence (or composition) of the Collatz function and its reduced variant, which serves as an analog to the inverse function, to scrutinize the validity of the Collatz conjecture. To substantiate this conjecture, we incorporate binary strings into an algebraic formula that captures the essence of the Collatz sequence. By this means, we transform discrete powers of 2 into continuous counterparts, ultimately culminating in the smallest natural number, 1. Consequently, the sequence generated through infinite iterations of the Collatz function emerges as an eventually periodic sequence, thereby validating an enduring 87-year-old conjecture.
Keywords
binary string; full binary directed tree; composite function; Collatz conjecture; ultimately periodic sequence
Subject
Computer Science and Mathematics, Algebra and Number Theory
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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