/Sandbox 23.12.2024/

On 17.10.2024, the article “A new inherent approach to solving the Collatz 3n+1 problem and its analogues” was published on the Academia.edu website [1]. The second link is for those who find it easier to read in Russian “Новый внутренне присущий подход к решению проблемы Коллатца 3n+1 и ее аналогов” [2].

The English version was originally intended for the arXiv preprint platform, but they suggested first publishing in a peer-reviewed mathematical journal. Attempts to access other platforms HAL, Qeios, and ResearchGate were met with the requirement for affiliation, which an independent researcher does not have.

The process took almost two months — more than the research itself from idea to text. As a result, the article ended up on the Academia site, which is free from “face control”. I think it will be useful for anyone interested in the Collatz conjecture to read it. Exclusively for Habr, this short text summarizing the content and meaning of the publication.

State of affairs

The Collatz conjecture (3n+1 problem) was first proposed over 80 years ago. It is formulated simply: take any natural number, if it is even, divide by 2 until it is divisible, and if it is odd, multiply by 3 and add 1, and repeat the same actions on the resulting number. Whatever the initial number we take, the result is always 1. The conjecture has not been proven or disproved, despite a huge number of works and the use of a wide variety of approaches.

At the same time, ideas about the object under study “3n+1” and its analogues “an±b” were usually limited to the judgment that this numerical sequence either converges (then the conjecture is true) or diverges or loops (then the conjecture is false). A deeper understanding was not offered. All the main more or less reliably established facts (about the convergence of 3n+1, about the existence of non-trivial cycles) were obtained empirically, by calculation.

The prolonged absence of a solution to a seemingly simple problem from the professional mathematical community has led to the popularity of the conjecture and a mass of amateur “proofs”. This state is extremely uncomfortable for everyone, not to mention the gigantic amount of wasted resources of all kinds.

What's changed

The article proposes a new approach, which consists in imposing the recursive Collatz algorithm on the table of expansion of all natural numbers in powers of two. It is surprising that mathematicians have passed by such an obvious idea. The problem was immediately simplified, since this construction guarantees the convergence of the algorithm under the condition that the network of all numbers is simply connected with a single root.

But this condition can be violated for a number of reasons: absence of a root, extra roots, cycles, network decay. The article describes methods for diagnosing these violations. After which only a part of the algorithms remained candidates for Collatz convergence. Candidates, because the last reason (network decay) is not so obvious. In the simplest case 1n+1 single-connectivity is easy to prove, but in the case 3n+1 and further it is really difficult.

However, here too a ray of light appeared. It became clear that the final proof should be by recurrence with completion or external to the Collatz algorithm. The author was saved from its painful search by discovering an article where just this kind of proof (correct, in my opinion) was presented by independent researcher Leszek Mazurek in May 2021 [3].

Those interested in details are referred to my article. Just 15 pages of text written intentionally understandable to everyone, with a minimum of formulas, breaking the tradition of not leaving any traces by which the reader could guess how the mathematical result was obtained.

In conclusion, self-quote: «To summarize: with the Collatz conjecture, if not everything, then a lot is already fairly clear. I guess it is time to rethink both the unsolved problem itself and the long, tangled history of attempts to solve it.»

[1] https://www.academia.edu/124810891

+ https://doi.org/10.5281/zenodo.14632983 (17.01.2025)

[2] https://www.academia.edu/126375432

+ https://drive.google.com/file/d/1Ntibbx8nFB7YDslLEF5RY7bZ40FfLdlO/view?usp=sharing

[3] https://www.researchgate.net/publication/351347153

/Update 25.12.2024/

Warning: this is not PR, but a necessary measure. I decided to give a partial “insider” (but “insight” would also work) from my article. At the request of a user who doubts whether to release my material (above) from the Sandbox.

On Habr you can find several publications devoted to cycles in algorithms similar to 3n+1. An example is Tzimie's article “Let's look at the Collatz conjecture” (https://habr.com/ru/articles/717094/), illustrated with beautiful fractals.

The scientific article “A new inherent approach to solving the Collatz 3n+1 problem and its analogues” [1, 2] contains an exhaustive explanation of the causes of cycles, what types they are, and why they are rare. For Tzimie, cycles are something computable but incomprehensible. In fact, they are just a manifestation of integer solutions to simple systems of linear equations (for a quick confirmation, see Addendum 30.09.2024 to said article).

So, there is nothing “mystical” in the Collatz cycles. I hope I shed light (of truth) on a question that intrigued many. And that's not all, the full version has a lot more interesting stuff (links are provided). In my not too subjective opinion, this turned out to be a must-read article for both professionals and amateurs of the Collatz conjecture.

The Collatz conjecture as a fail of world math