Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture

Year 2020, , 120 - 129, 29.12.2020

Ramon Carbó Dorca

https://doi.org/10.33187/jmsm.776898

Cited By: 2

Abstract

This study is based on the trivial transcription of the vertices of a Boolean \textit{N}-Dimensional Hypercube $\textbf{H}_{N} $ into a subset $\mathbb{S}_{N}$ of the decimal natural numbers $\mathbb{N}.$ Such straightforward mathematical manipulation permits to achieve a recursive construction of the whole set $\mathbb{N}.$ In this proposed scheme, the Mersenne numbers act as upper bounds of the iterative building of $\mathbb{S}_{N}$. The paper begins with a general description of the Collatz or $\left(3x+1\right)$ algorithm presented in the $\mathbb{S}_{N} \subset \mathbb{N}$ iterative environment. Application of a defined \textit{ad hoc} Collatz operator to the Boolean Hypercube recursive partition of $\mathbb{N}$, permits to find some hints of the behavior of natural numbers under the $\left(3x+1\right)$ algorithm, and finally to provide a scheme of the Collatz conjecture partial resolution by induction.

Keywords

Boolean Hypercubes, Recursive Construction of Natural Numbers, Mersenne Numbers, Mersenne Twins, Collatz Conjecture, (3x+1) Conjecture, Collatz Algorithm, Collatz Operator

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