Proving the correctness of the Collatz hypothesis
DOI:
https://doi.org/10.52754/16948645_2025_4(1)_8Keywords:
natural numbers; odd number potential; mathematical induction; sequence analysis; open problemAbstract
Proof of the correctness of the Collatz conjecture is topical research, as it represents one of the many unsolved problems in mathematics. Understanding the properties of this sequence has important implications for other areas of mathematics, such as number theory or graph theory. The aim of the study was to prove the Collatz hypothesis as a theorem. The research methodology included the analysis of numerical sequences, the use of mathematical induction, recursive, combinatorial methods and computer modelling. The study analysed the properties of sequences generated by the Collatz hypothesis, particularly their recursive properties. The study determined that each odd number has a unique “potential” that affects the behaviour of the sequence. The correlation between even and odd numbers in the context of the hypothesis, as well as the influence of division and multiplication operations on the change of number sequences, are investigated. The results of the study showed that sequences according to the Collatz hypothesis have specific patterns that can be used to develop effective approaches to their proof. The study also determined that the operations of dividing by 2 multiplying by 3 and adding 1 have a systemic effect on the development of the sequence. The results of the study showed that the proposed method of studying sequences helped to determine the correct location of numbers in an infinite sequence of natural numbers N and other groups of numbers. The main difference of the proposed approach is the introduction of the concept of “potential of an odd number” and “blocks of numbers” related to this odd number. The potential of an odd number was a property of numbers that confirmed the hypothesis and was used to call the Collatz problem a theorem. The practical significance of the study lies in the possibility of applying new methods of analysing numerical sequences in computer science, cryptography and other fields requiring optimisation of computing processes
References
Zelensky OV, Dynych AY, Darmosiuk VM, Lobach RV. Properties of possible counterfactuals to the Collatz hypothesis. In: Proceedings of the XI international scientific and practical conference “Methodological and attitudinal principles of classical science”. Stockholm: InterSci; 2023. P. 27–31.
Kosobutskyy P, Karkulovskyy V. Recurrence and structuring of sequences of transformations 3n+1 as arguments for confirmation of the Collatz hypothesis. Comput Des Syst Theor Pract. 2023;5(1):28–33. DOI: 10.23939/cds2023.01.028
Kmita SV. The Collatz hypothesis. In: Collection of scientific papers of the IX All-Ukrainian scientific and practical conference of cadets and students “Mathematics that surrounds us: Past, present, future”. Lviv: Lviv State University of Life Safety; 2022. P. 143–5.
Kosobutskyy P, Rebot D. Collatz conjecture 3n±1 as a newton binomial problem. Comput Des Syst Theor Pract. 2023;5(1):137–45. DOI: 10.23939/cds2023.01.137
Leshchenko O. Collatz hypothesis – the simplest unsolved problem of mathematics. Khortytsia Read. 2023;6:82–7.
Dyachenko E. A proof of the Collatz conjecture and connection of intervals based on “tetrada”. Genève: Zenodo; 2020. DOI: 10.5281/zenodo.4013334
Stérin T. Binary expression of ancestors in the Collatz graph. In: Schmitz S, Potapov I, editors. International conference on reachability problems. Cham: Springer; 2020. P. 115–30. DOI: 10.1007/978-3-030-61739-4_8
Nwogugu MC. On the Collatz conjecture. SSRN. 2022. DOI: 10.2139/ssrn.4170807
Diedrich E. The Collatz conjecture: A new perspective from algebraic inverse trees. Int J Pure Appl Math Res. 2023;4:34–79. DOI: 10.20944/preprints202310.0773.v13
Izadi F. Complete proof of the Collatz conjecture. ArXiv. 2021. DOI: 10.48550/arXiv.2101.06107
Danesi M. Ingenuity. In: Danesi M, editor. Poetic logic and the origins of the mathematical imagination. Cham: Springer; 2023. P. 33–66. DOI: 10.1007/978-3-031-31582-4_2
Trocado A, Dos Santos JM, Lavicza Z. Developing computational thinking in Portuguese mathematics curricula with Collatz conjecture. In: The 27th Asian technology conference in mathematics. Singapore; 2022. P. 363–72.
Fan C, Ding Q. Design and geometric control of polynomial chaotic maps with any desired positive Lyapunov exponents. Chaos Solitons Fractals. 2023;169:113258. DOI: 10.1016/j.chaos.2023.113258
Rasool M, Belhaouari SB. From Collatz conjecture to chaos and hash function. Chaos Solitons Fractals. 2023;176:114103. DOI: 10.1016/j.chaos.2023.114103
Fabiano N, Mirkov N, Mitrović ZD, Radenović S. Collatz hypothesis and Kurepa’s conjecture. In: Debnath P, Srivastava HM, Chakraborty K, Kumam P, editors. Advances in number theory and applied analysis. Singapore: World Scientific Publishing; 2023. P. 31–50. DOI: 10.1142/9789811272608_0003
Koch C, Sultanow E, Cox S. Divisions by two in Collatz sequences: A data science approach. Int J Pure Math Sci. 2020;21:3–24.
Tadesse H. A proof of the Collatz conjecture [Internet]. OSF Preprints; 2023. DOI: 10.31219/osf.io/e8x2w
Machado JA, Galhano A, Cao Labora D. A clustering perspective of the Collatz conjecture. Mathematics. 2021;9(4):314. DOI: 10.3390/math9040314
Gurbaxani BM. An engineering and statistical look at the Collatz (3n+1) conjecture. ArXiv. 2021. DOI: 10.48550/arXiv.2103.15554
Rahn A, Sultanow E, Henkel M, Ghosh S, Aberkane IJ. An algorithm for linearizing the Collatz convergence. Mathematics. 2021;9(16):1898. DOI: 10.3390/math9161898
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