natural numbers

For the first time, a statistical model of a pseudo-random number generator (PRNG) with the Collatz transformation function is constructed and investigated in this paper. The PRNG is implemented in the Python statistical programming environment, and the function is obtained using the inverse transformation method. It is established that the probability integral function takes the form of a transcendental polynomial of quadratic nature, within which the range of PRNG values is justified.

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 power transformation of Newton's binomial forms two equal 3n±1 algorithms for transformations of numbers n belongs to N, each of which have one infinite cycle with a unit lower limit of oscillations. It is shown that in the reverse direction, the Kollatz sequence is formed by the lower limits of the corresponding cycles, and the last element goes to a multiple of three odd numbers.

It is shown that: 1. The sequence {2⁰,2¹, 2², 2³, 2⁴, 2⁵, 2⁶, 2⁷,2⁸,...} that forms the main graph m=1 of Collatz is related to the power transformation of Newton's binomial (1+1)^ξ, ξ=0, 1, 2, 3,... 2. The main K_main and side m >1 graphs and their corresponding sequences {K_main } and {K_m } are related by the relation {K_m }=m⋅{K_main }. 3.

It is shown that infinites of the subsequence of odd numbers is not a counterargument of the violation of the Collatz hypothesis, but a universal characteristic of transformations of natural numbers by the 3n + 1 algorithm. A recurrent relationship is established between the parameters of the sequence of Collatz transformations of an arbitrary pair of natural numbers n and 2n.