Jacobsthal numbers

The work is devoted to the study of dynamical systems with point attractors by the recurrent method of transforming discrete data from the set of natural numbers, in the direction of increasing powers of two (direct Jacobsthal problem) and in the opposite direction (reverse Collatz problem). The idea of splitting the set N into separate non-overlapping subsets by Jacobsthal transformation of numbers was also expressed for the first time.

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.

In this work justified incorrectness of the algorithm proposed in the publication "M. Remer.[A Comparative Analysis of the New -3(-n) - 1 Remer Conjecture and a Proof of the 3n + 1 Collatz Conjecture. Journal of Applied Mathematics and Physics. Vol.11 No.8, August 2023"] in terms of the Collatz conjecture. And also that the transformation -3(-n) - 1 is not equivalent to Collatz's conjecture on the natural numbers 3n + 1. The obtained results can be used in further studies of the Collatz hypothesis

The work shows that the task is the problem C_3q±​1=3q±1 conjecture positive integers q>1in the reverse direction n→0 of the branching of the Jacobsthal tree, according to the rules of transformations of recurrent Jacobsthal numbers. For the first time, the Collatz problem is analyzed from the point of view of the increase in information entropy after the passage of the so-called fusion points (nodes) on the polynomials θ*2ⁿ by the Сollatz trajectories.