Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture

Ramon Carbó Dorca

doi:10.33187/jmsm.776898

EN

Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture

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

References

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Details

Primary Language

English

Subjects

Mathematical Sciences

Journal Section

Research Article

Authors

Ramon Carbó Dorca ^*
Spain

Publication Date

December 29, 2020

Submission Date

August 4, 2020

Acceptance Date

December 18, 2020

Published in Issue

Year 2020 Volume: 3 Number: 3

DOI

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

IZ

https://izlik.org/JA54LA58AJ

APA

Carbó Dorca, R. (2020). Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture. Journal of Mathematical Sciences and Modelling, 3(3), 120-129. https://doi.org/10.33187/jmsm.776898

AMA

1.Carbó Dorca R. Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture. Journal of Mathematical Sciences and Modelling. 2020;3(3):120-129. doi:10.33187/jmsm.776898

Chicago

Carbó Dorca, Ramon. 2020. “Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture”. Journal of Mathematical Sciences and Modelling 3 (3): 120-29. https://doi.org/10.33187/jmsm.776898.

EndNote

Carbó Dorca R (December 1, 2020) Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture. Journal of Mathematical Sciences and Modelling 3 3 120–129.

IEEE

[1]R. Carbó Dorca, “Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture”, Journal of Mathematical Sciences and Modelling, vol. 3, no. 3, pp. 120–129, Dec. 2020, doi: 10.33187/jmsm.776898.

ISNAD

Carbó Dorca, Ramon. “Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture”. Journal of Mathematical Sciences and Modelling 3/3 (December 1, 2020): 120-129. https://doi.org/10.33187/jmsm.776898.

JAMA

1.Carbó Dorca R. Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture. Journal of Mathematical Sciences and Modelling. 2020;3:120–129.

MLA

Carbó Dorca, Ramon. “Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture”. Journal of Mathematical Sciences and Modelling, vol. 3, no. 3, Dec. 2020, pp. 120-9, doi:10.33187/jmsm.776898.

Vancouver

1.Ramon Carbó Dorca. Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture. Journal of Mathematical Sciences and Modelling. 2020 Dec. 1;3(3):120-9. doi:10.33187/jmsm.776898