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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)
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.v1 Feng, J. Collatz Conjecture Is Analogous to an Inverse Function of Natural Number. Preprints 2024, 2024050310. https://doi.org/10.20944/preprints202405.0310.v1
Abstract
We propose a full binary directed tree to represent the set of natural numbers and further divide the set into three sets: pure odd, pure even, and mixed numbers. We utilize a binary string to represent a natural number and demonstrate the composite procedure of odd-number and even-number functions. We analyze the sequence of iteration (or composite) of the Collatz function and reduced Collatz function analog to the inverse function in order to test the Collatz conjecture. We do this by using the parity of a natural number. In order to prove the conjecture, we provide tabular and binary strings to the algebraic formula that states the Collatz sequence. Ultimately, we can convert discrete powers of 2 into continuous powers of 2, ultimately arrive at the smallest natural, 1. If any natural number is the beginning value, the sequence produced by the infinite iterations of the Collatz function becomes the eventually periodic sequence, proving an 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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