Boolean Hypercubes, Classification of Natural Numbers, and the Collatz Conjecture

Yıl 2022, Cilt: 5 Sayı: 3, 80 - 91, 01.12.2022

Ramon Carbó Dorca Carlos Perelman

https://doi.org/10.33187/jmsm.972781

Öz

Using simple arguments derived from the Boolean hypercube configuration, the structure of natural spaces, and the recursive exponential generation of the set of natural numbers, a linear classification of the natural numbers is presented. The definition of a pseudolinear Collatz operator, the description of the set of powers of $2$, and the construction of the natural numbers via this power set might heuristically prove the Collatz conjecture from an empirical point of view.

Anahtar Kelimeler

Boolean Hypercubes, Collatz conjecture, Collatz vector spaces, Mersenne numbers, Natural hypercubes

Kaynakça

• [1] R. Carbo-Dorca, Boolean Hypercubes, Mersenne numbers and the Collatz conjecture, J. Math. Sci. Mod., 3 (2020), 120-129.
• [2] The Collatz conjecture. Wikipedia, https://en.wikipedia.org/wiki/Collatz conjecture “What is the Importance of the Collatz Conjecture?” https://math.stackexchange.com/questions/2694/what-is-the-importanceof-the-collatz-conjecture.
• [3] H. Nowak, Collatz conjecture and emergent properties, https://www.youtube.com/watch?v=QrzcHhBQ2b0.
• [4] J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92(1) (1985), 323.
• [5] J. C. Lagarias, A. Weiss, The 3x+1 problem: two stochastic models, Ann. Applied Prob. 2 (1992), 329-361.
• [6] I. Korec, A density estimate for the 3x+1 problem, Math. Slovaca, 44 (1994), 85-89.
• [7] G. J. Wirsching, The Dynamical System Generated by the 3n+1 Function, Lecture Notes in Math. 1681, Springer-Verlag, New York, 1998.
• [8] J. C. Lagarias, K. Soundararajan, Benford’s law for the 3x+1 function, J. London Math. Soc., 74 (2006), 289-303.
• [9] J. C. Lagarias, ed.; The ultimate challenge: the 3x+1 problem. Providence, R. I.: Amer. Math. Soc., (2010).
• [10] F. Oan, J. P. Draayer, A polynomial approach to the Collatz conjecture, arXiv.org: 1905.08462 [math. NT] (2019).
• [11] F. Izadi, A new approach on proving Collatz conjecture, Journal of Mathematics, Hindawi, 2019 (2019), Article ID: 6129836, 12 pages.
• [12] D. Barina, Convergence verification of the Collatz problem, The Journal of Supercomputing, 77(3) (2021), 2681-2688.
• [13] T. Tao, Almost all orbits of the Collatz map attain almost bounded values, In Forum of Mathematics, Pi (Vol. 10). Cambridge University Press, 2022.
• [14] J. A. T. Machado, A. Galhano, D. Cao Labora, A clustering perspective of the Collatz conjecture, Mathematics, 9 (2021), 314-328.
• [15] F. Izadi, Complete proof of Collatz’s conjectures, arXiv:2101.06107v4 [math. GM], (2021).
• [16] B. M. Gurbaxani, An engineering and statistical look at the Collatz (3n+1) conjecture ResearchGate Preprint, 14 March (2021).
• [17] B. B. Stefanov, Two-parameter generalization of the Collatz function characterization of terminal cycles and empirical results, Online Mathematics OMJ, 03(01) (2021), 19-25.
• [18] L.-O. Pochon, A. Favre, La suite de Syracuse, un monde de conjectures, (2021) ffhal01593181v3f.
• [19] C. Castro Perelman, R. Carb´o-Dorca, The Collatz conjecture and the quantum mechanical harmonic oscillator, J. Math. Chem. 60 (2022), 145-160.
• [20] N. Fabiano, N. Mirkov, S. Radenovic, Collatz hypothesis and Planck’s black body radiation, J. Siberian Fed. Univ. Mathematics & Physics, 2 (2021), 1-5.
• [21] A. Rahn, E. Sultanov, M. Henkel, S. Ghosh, I. J. Aberkane, An algorithm for linearizing the Collatz convergence, Mathematics, 9 (2021), 1898-1930.
• [22] J. Kleinnijenhuis, A. M. Kleinnijenhuis, Pruning the binary tree, proving the Collatz conjecture, ResearchGate Preprint (no file attached), August (2020).
• [23] M. R. Schwob, P. Shiue, R. Venkat, Novel theorems and algorithms relating to the Collatz conjecture, Int. J. Math. Math. Sci., 2021 (2021), Article ID: 5754439, 10 pages, (2021).
• [24] https://www.mersenne.org/primes/
• [25] R. Carb´o-Dorca, Cantor-like infinity sequences and G¨odel-like incompleteness revealed by means of Mersenne infinite dimensional Boolean Hypercube concatenation, J. Math. Chem., 58 (2020), 1-5.
• [26] R. Carbo-Dorca, About Erd¨os discrepancy conjecture, J. Math. Chem., 54 (2016), 657-660.
• [27] R. Carbo-Dorca, N-dimensional Boolean Hypercubes and the Goldbach conjecture, J. Math. Chem., 54 (2016), 1213-1220.
• [28] 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.
• [29] R. Carbo-Dorca, Boolean Hypercubes and the structure of vector Spaces, J. Math. Sci. Mod., 1 (2018), 1-14.
• [30] R. Carbo-Dorca, Role of the structure of Boolean Hypercubes when used as vectors in natural (Boolean) vector semi spaces, J. Math. Chem., 57 (2019), 697-700.
• [31] R. Carbo, E. Besal´u, Definition, mathematical examples and quantum chemical applications of nested summation symbols and logical Kronecker deltas, Computers & Chemistry 18 (1994), 117-126.
• [32] E. Besalu, R. Carbo, Definition and quantum chemical applications of nested summation symbols and logical Kronecker deltas: Pedagogical Artificial intelligence devices for formulae writing, sequential programming and automatic parallel implementation, J. Math. Chem., 18 (1995), 37-72.
• [33] R. Carbo-Dorca, Logical Kronecker delta deconstruction of the absolute value function and the treatment of absolute deviations, J. Math. Chem., 49 (2011), 619-624.
• [34] R. Carbo-Dorca, Inward matrix products: Extensions and applications to quantum mechanical foundations of QSAR, J. Mol. Struct. Teochem, 537 (2001), 41-54.
• [35] R. Carbo-Dorca, Inward Matrix product algebra and calculus as tools to construct space-time frames of arbitrary dimensions, J. Math. Chem., 30 (2001), 227-245.