Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture - Ramon Carbó Dorca (Journal of Mathematical Sciences and Modelling, 2020)
This study is based on the trivial transcription of the vertices of a Boolean N-Dimensional Hypercube $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 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.