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Using Full Binary Directed Tree Proof the Collatz Conjecture
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: Received: 7 May 2024 / Approved: 8 May 2024 / Online: 8 May 2024 (14:21:59 CEST)
How to cite: Feng, J. Using Full Binary Directed Tree Proof the Collatz Conjecture. Preprints 2024, 2024050510. https://doi.org/10.20944/preprints202405.0510.v1 Feng, J. Using Full Binary Directed Tree Proof the Collatz Conjecture. Preprints 2024, 2024050510. https://doi.org/10.20944/preprints202405.0510.v1
Abstract
We use binary string to represent a natural number and show the composite procedure of odd-number and even-number functions, thus we propose full binary directed tree to represent the set of natural numbers and give another partition the natural number set to three sets: pure odd, pure even and mixed number. For the Collatz conjecture we make use of the parity of a natural number to analyse the sequence of iteration (or composite) of Collatz function and reduced Collatz function analog to the inverse function. We give tabular and binary strings to the algebra expression to state the sequence of Collatz, this is the key topic to proof the conjecture: the discrete powers of 2 can be changed to ultimately continuous powers of 2, finally get to pure even number and to the smallest number 1. The sequence created by the infinite iterations of the Collatz function becomes the ultimately periodic sequence if any natural number is the beginning value, proving the conjecture that has been held for 87 years.
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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