Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture

Authors : Ramon CARBÓ DORCA

Pages : 120-129

Doi:10.33187/jmsm.776898

View : 25 | Download : 14

Publication Date : 2020-12-29

Article Type : Research Paper

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 $\leftinsert ignore into journalissuearticles values(3x+1\right);$ 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 $\leftinsert ignore into journalissuearticles values(3x+1\right);$ 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