Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture
Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture
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,
___
- [1] J. C. Lagarias, The 3x+1 problem and its generalizations, Am. Math. Monthly, 92 (1985), 3-23.
- [2] J. C. Lagarias (Editor), The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010.
- [3] Collatz conjecture, available at https://en.wikipedia.org/wiki/
- [4] T. Tao, Almost all orbits of the collatz map attain almost bounded values, (2019), arXiv:1909.03562v2 [math.PR].
- [5] J. C. Lagarias, The 3x+1 problem: An Annotated Bibliography, I (1963-1999), (2009), arXiv:math/0309224v12 [math.NT].
- [6] J. C. Lagarias, The 3x+1 problem: An Annotated Bibliography, II (2000-2009), (2009), arXiv:math/0608208v5 [math.NT].
- [7] R. Carbo-Dorca, Boolean hypercubes and the structure of vector spaces, J. Math. Sci. Modelling, 1 (2018), 1-14.
- [8] R. Carbo-Dorca, Natural vector spaces, (Inward power and Minkowski norm of a natural vector, natural Boolean hypercubes) and Fermat’s last theorem, J. Math. Chem., 55 (2017), 914-940.
- [9] R. Carbo-Dorca, C. Munoz-Caro, A. Ni˜no, S. Reyes, Refinement of a generalized Fermat’s last theorem conjecture in natural vector spaces, J. Math. Chem., 55 (2017), 1869-1877.
- [10] R. Carbo-Dorca, Cantor-like infinity sequences and Godel-like incompleteness revealed by means of Mersenne infinite dimensional Boolean hypercube concatenation, J. Math. Chem., 58 (2020), 1-5.
- [11] R. Carbo-Dorca, Fuzzy sets and Boolean tagged sets, J. Math. Chem., 22 (1997), 143-147.
- [12] R. Carbo, B. Calabuig, Molecular similarity and quantum chemistry, M. A. Johnson, G. M. Maggiora (editors) Chapter 6 in Concepts and Applications of Molecular Similarity, John Wiley & Sons Inc., New York, 1990, pp. 147-171.
- [13] R. Carbo, B. Calabuig, Molecular Quantum Similarity Measures and N-Dimensional Representation of Quantum Objects II. Practical Applications (3F-Propanol conformer taxonomy among other examples), Intl. J. Quant. Chem., 42 (1992), 1695-1709.
- [14] R. Carbo-Dorca, About Erd¨os discrepancy conjecture, J. Math. Chem., 54 (2016), 657-660.
- [15] R. Carbo-Dorca, N-dimensional Boolean hypercubes and the Goldbach conjecture, J. Math. Chem., 54 (2016), 1213-1220.
- [16] R. Carbo-Dorca, A study on Goldbach conjecture, J. Math. Chem., 54 (2016), 1798-1809.
- [17] R. Carbo-Dorca, Boolean hypercubes as time representation holders, J. Math. Chem., 56 (2018), 1349-1352.
- [18] R. Carbo-Dorca, DNA, unnatural base pairs and hypercubes, J. Math. Chem., 56 (2018), 1353-1356.
- [19] R. Carbo-Dorca, Transformation of Boolean hypercube vertices into unit interval elements: QSPR workout consequences, J. Math. Chem., 57 (2019), 694-696.
- [20] R. Carbo-Dorca, Role of the structure of Boolean hypercubes when used as vectors in natural (Boolean) vector semispaces, J. Math. Chem., 57 (2019), 697-700.
- [21] R. Carbo-Dorca, T. Chakraborty, Hypercubes defined on n-ary sets, the Erd¨os-Faber-Lov´asz conjecture on graph coloring, and the polypeptides and RNA description spaces, J. Math. Chem., 57 (2019), 2182-2194.
- [22] J. Chang, R. Carbo-Dorca, Fuzzy hypercubes and their time-like evolution, J. Math. Chem., 58 (2020), 1337–1344.
- [23] K. Balasubramanian, Combinatorial multinomial generators for colorings of 4D-hypercubes and their applications, J. Math. Chem., 56 (2018), 2707-2723.
- [24] K. Balasubramanian, Computational multinomial combinatorics for colorings of 5D-hypercubes for all irreducible representations and applications, J. Math. Chem., 57 (2018), 655-689.
- [25] https://www.mersenne.org/primes/
- [26] A.V. Kontorovich, J. C. Lagarias, Stochastic models for the 3x+1 and 5x+1 problems, (2009), arXiv:0910.1944v1 [math.NT].
- [27] http://www.ericr.nl/wondrous/
- [28] W. Ren A new approach on proving collatz conjecture, Hindawi J. Math., (2019), Article ID 6129836, 1-12.
- ISSN: 2636-8692
- Yayın Aralığı: Yılda 3 Sayı
- Başlangıç: 2018
- Yayıncı: Mahmut AKYİĞİT
Sayıdaki Diğer Makaleler
On Bicomplex Jacobsthal-Lucas Numbers
Early Childhood Children in COVID-19 Quarantine Days and Multiple Correspondence Analysis
Nihat TOPAÇ, Musa BARDAK, Seda BAĞDATLI KALKAN, Murat KİRİSCİ
Boolean Hypercubes, Mersenne Numbers, and the Collatz Conjecture
Fitting an Epidemiological Model to Transmission Dynamics of COVID-19
Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation