A new branching tree model has been proposed for the first time in the direction of increasing degree 2n (merging in the reverse direction), which coincides with the direction of increasing total stopping time. It has been shown that each time corresponds to a sequence of individual numbers n(tst)→∞, the volume of which increases with time. Thus, it is proven that each time corresponds to a finite number of Collatz sequences of the same length. The reason for the formation of a histogram or spectrum tst(q) with two peaks has been established. It is shown that the double structure is formed by the regularities of Jacobsthal recurrence numbers at the nodes of the sequences. It has been established that the graph tst(q) with the numbers of active nodes in semi-logarithmic coordinates tst, logm(p) appears as a straight line, while the graph for the numbers of inactive nodes appears as a scattered spectrum. Based on the established statistical regularities tst(q), a new recurrent model of trivial cycles is proposed.
[1] R.Terras. A stopping time problem on the positive integers. Acta Arith. 30: 241–252,197
[2] C. Lagarias, The (3x+1)–problem and its generalizations, American Mathematical Monthly 92 (1985), 3–23.
[3] K. A. Borovkov and D. Pfeifer, Estimates for the Syracuse Problem via a probabilistic model, Theory of Probability and its Applications 45, N2 (2000), 300–310.
[4] G. J. Wirsching, The Dynamical System generated by the (3x+ 1)–function, Lecture Notes in Mathematics, N1681, Springer–Verlag, Berlin, 1998, 158p.
[5] B.Gurbaxani. An Engineering and Statistical Look at the Collatz (3n + 1) Conjecture. arXiv preprint arXiv:2103.15554
[6] M. Rasool, S.Belhaouari. From Collatz Conjecture to chaos and hash function. Chaos, Solitons and Fractals 176 (2023) 114103, 2023. http://creativecommons.org/licenses/by/4.0/
[7] A. Grubiy. Automation implementations of the process of generating Collatz sequence. Vol.48, pp.108-116,2012
[8] Y. Sinai. Statistical (3x+ 1) problem, Dedicated to the memory of Jurgen K. Moser. Communications InPure & Applied Math., 56(7), 1016–1028,2003ю
[9] T. Tao. Almost all orbits of the Collatz map attain almost bounded values. Forum of Mathematics, Pi,Volume (10),2022.
[10] http://en.wikipedia.org/wiki/File:CollatzStatistic100million.png
[11] C. AllenMc. Histogram of total stopping times for the numbers 1 to 100 million (2013). Link: https://en.wikipedia.org/wiki/Collatz_conjecture#/media/File: CollatzStatistic100million.png
[12] Thomas e Silva. Computational Verification of the 3x+1 conjecture, Universidade de Aveiro (2015). Link: http://sweet.ua.pt/tos/3x+1.html
[13] U. Rinat. Collatz Conjecture: calculation in reverse with JavaScript. https://blog.rinatussenov.com/collatz-conjecture-calculation-in-reverse-with-javascript-a768fab10425
[14] J. Miller. Reversing the Collatz Conjecture Linearly. https://medium.com/@jordan.kay/reversing-the-collatz-conjecture-linearly...
[15] N. Fabiano, Z.Mitrovic, N.Mirkov, S.Radenović. A discussion on two old standing number theory problems: Collatz hypothesis, together with its relation to Planck’s black body radiation, and Kurepa’s conjecture on left factorial function Chapter 1. October 2022h. ttps://www.researchgate.net/publication/364284245
[16] P. Kosobutskyy. The Collatz problem as a reverse problem on a graph tree formed from Q*2^n (Q=1,3,5,7,…) Jacobsthal-type numbers .arXiv:2306.14635v1
[17] P. Kosobutskyy. Comment from article ”Two different scenarios when the Collatz Conjecture fails”. General Letters in Mathematics. 2022. Vol. 12, iss. 4. P. 179–182.
[18] P. Kosobutskyy, D. Rebot. Collatz conjecture 3n ± 1 as a Newton Binomial Problem. Computer Design Systems. Theory and Practice, Vol. 5, No. 1, 2023.рр.137-145
[19] P. Kosobutskyy, Yedyharova A., Slobodzyan T. From Newton's binomial and Pascal’s triangle to Collatz's problem. Computer Design Systems. Theory and Practice , Vol. 5, No 1, 2023.рр.121-127
[20] P. Kosobutskyy, Karkulovskyy V. Recurrence and structuring of sequences of transformations 3n +1 as arguments for confirmation of the Collatz hypothesis. Computer Design Systems. Theory and Practice. Vol. 5, No. 1, 2023.рр.28-33
[21] J. Choi. Ternary Modified Collatz Sequences And Jacobsthal Numbers. Journal of Integer Sequences, Vol. 19 (2016), Article 16.7.5
[22] Sloan's On-Line Encyclopedia of Integer Sequences (OEIS, http://oeis.org/).