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Collatz conjecture: an order isomorphic recursive machine
Version 1
: Received: 30 March 2022 / Approved: 31 March 2022 / Online: 31 March 2022 (08:03:45 CEST)
Version 2 : Received: 29 February 2024 / Approved: 1 March 2024 / Online: 1 March 2024 (10:32:19 CET)
Version 2 : Received: 29 February 2024 / Approved: 1 March 2024 / Online: 1 March 2024 (10:32:19 CET)
How to cite: Williams, M. Collatz conjecture: an order isomorphic recursive machine. Preprints 2022, 2022030401. https://doi.org/10.20944/preprints202203.0401.v2 Williams, M. Collatz conjecture: an order isomorphic recursive machine. Preprints 2022, 2022030401. https://doi.org/10.20944/preprints202203.0401.v2
Abstract
Collatz conjecture (3x+1 problem) is a natural phenomenon in set theory that may be reconstructed using known combinatorics and order theory. Construction begins by selecting a specific order isomorphism with a bijectional order-embedding. Mapping by 3x+1 induces a unique property to only members of the mapped embedding. Under selective congruence from recursive division by two, that complex controls sequence behavior. This demonstration uses an order isomorphism consisting of two linear orders: the (1) odd positive integers and the always odd (2) Rule 50 Jacobsthal numbers, as the embedding. The argument proceeds by cardinality. When the order isomorphism is mapped under 3x+1, all Rule 50 Jacobsthal numbers are mapped to all the powers of four. This one-to-one correspondence guarantees that every odd integer is assigned to a Rule 50 Jacobsthal number, and subsequently, to a power of four. To align this construction with the conjecture, expand the set to all positive integers by showing that: (1) the powers of four embed the powers of two, in a natural way, and (2) the unique factorization of any positive even integer, not congruent to a power of two, is simply an odd integer, once the factorized power of two is divided out. In other words, if the initial choice for a positive integer is not congruent to a power of two, then recursion continues until a Rule 50 Jacobsthal number (guaranteed by cardinality) is attained. Since this value mapped by 3x+1 is always a power of four, repeated division by two will always send the sequence to one. Because this same process occurs for any choice of positive integer, Collatz conjecture is true.
Keywords
Collatz conjecture; order isomorphism; embedding; Rule 50; Jacobsthal numbers; congruence
Subject
Computer Science and Mathematics, Applied Mathematics
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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