Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

# Collatz Conjecture: An Order Machine

Version 1 : Received: 30 March 2022 / Approved: 31 March 2022 / Online: 31 March 2022 (08:03:45 CEST)

How to cite: Williams, M. Collatz Conjecture: An Order Machine. Preprints 2022, 2022030401. https://doi.org/10.20944/preprints202203.0401.v1 Williams, M. Collatz Conjecture: An Order Machine. Preprints 2022, 2022030401. https://doi.org/10.20944/preprints202203.0401.v1

## Abstract

Collatz conjecture (3n+1 problem) is an application of Cantor's isomorphism theorem (Cantor-Bernstein) under recursion. The set of 3n+1 for all odd positive integers n, is an order isomorphism for (odd X, 3X+1). The other (odd X, 3X+1) linear order has been discovered as a bijective order-embedding, with values congruent to powers of four. This is demonstrated using a binomial series as a set rule, then showing the isomorphic structure, mapping, and cardinality of those sets. Collatz conjecture is representative of an order machine for congruence to powers of two. If an initial value is not congruent to a power of two, then the iterative program operates the (odd X, 3X+1) order isomorphism until an embedded value is attained. Since this value is a power of four, repeated division by two tends the sequence to one. Because this same process occurs, regardless of the initial choice for a positive integer, Collatz conjecture is true.

## Keywords

Collatz conjecture; hailstone sequences; Cantor isomorphism; order embedding; Jacobsthal numbers; congruence

## Subject

Computer Science and Mathematics, Applied Mathematics