Числа Якобсталя в задачі Коллатца з точки зору фізика / Jacobsthal numbers in the Collatz problem from the physicist's point of view

Abstract: У доповіді розглядається ідея про те, що траєкторіями перетворень Коллатца є гілки дерева Якобсталя, яке формується на основі закономірностей перетворень рекурентних чисел Якобсталя, в напрямку зростання степеня $Q \cdot 2^n$, реверсному до напрямку перетворення Коллатца. Показано, що правила перетворень чисел Якобсталя, що формально перенесені в задачу Коллатца, однозначні лише в напрямку формування дерева. Модель дерева Якобсталя узагальнено на модель перетворення загального типу $a x \pm 1$, де $a = 1, 2, 3, \ldots$.

In the talk, we consider an idea that trajectories of Collatz transformations are branches of the Jacobsthal tree that is formed on the base of transformations of recurrent Jacobsthal numbers in the direction of increasing of power $Q \cdot 2^n$ that is reversed to direction of Collatz transformation. We show that rules of transformations of Jacobsthal numbers that are formally transferred to Collatz problem are unambiguous in the direction of formation of the tree only. The model of the Jacobsthal tree is generalized to the model of general-type transformation $a x \pm 1$, where $a = 1, 2, 3, \ldots$.

Ukrainiannumber theory

Audience: advanced learners

( paper )

---

Семінар з фрактального аналізу / Fractal analysis seminar

Series comments: Weekly research seminar on fractal analysis (online)

Topics:

• theory of fractals (fractal geometry and fractal analysis)
• Hausdorff–Besicovitch dimension, techniques and methods for its calculation and estimation
• functions and transformations preserving fractal (Hausdorff–Besicovitch, entropic, box-counting, packing, etc.) dimension
• sets of metric spaces that are essential for functions, sets, and dynamical systems
• self-similar, self-affine properties of mathematical objects
• systems of encoding for real numbers (numeral systems) and their applications
• metric number theory and metric theory of representations of numbers
• probabilistic number theory and probabilistic theory of representations of numbers
• measures supported on fractals, particularly singular measures and probability distributions
• nowhere monotonic and nowhere differentiable functions, functions with fractal properties
• theory of groups determined by invariants of transformations preserving tails of representation of numbers, digit frequencies, etc.

The talks are mostly in Ukrainian but English is also acceptable

---