General Letters in Mathematics (GLM)

This work substantiates the essence of the erroneous conclusion of the author of the work ‘M. Ahmed, Two different scenarios when the Collatz Conjecture fails. General Letters in Mathematics. 2021‘ about the false of Collatz’s hypothesis. Keywords


Introduction
'The Collatz Conjecture is false.End of story', this is how the work ends [1].This is a bold conclusion and could not be ignored.This work shows that the conclusion made is hasty and here is why.Consider sequences of numbers {ℑ θ,n } ∞ n=0 , parametrized by an odd natural number θ whose terms are determined by the equality and group them by even and odd powers as where θ is the index of sequences of numbers θ • 2 n .Let's divide the subset of odd natural numbers into three groups, as shown in Fig. 1, and formulate the following properties of numbers (1.3).
Let's confirm properties 1 and 2 with concrete calculations: Shaded cells in Table 1 correspond to fractional values of numbers (1.3) So, if the index θ changes as θ 1,i = 1 + 6 • i fig. 1, and in calculations Table 1 θ 1 = 1, 7, then the numbers m θ,k take integer values, and the numbers p θ,l take fractional values.If the index θ changes as θ 5,i = 5 + 6 • i (fig.1), and in calculations Table 1 θ 5 = 5, 11, then the numbers m θ,k take fractional values, and the numbers p θ,l take integer values.If the index θ is a multiple of three (in Fig. 1), the series (θ 3 = 3 + 6 • i) and in calculations Table 1 θ 3 = 3, then both numbers m θ,k , p θ,l are fractional.Calculations Table 1 also confirm rule (1.9).
From (1.3) and properties 1 and 2, we formulate a system of equations: According to (1.11), every odd number θ generates sequences θ .12) Construction (1.12) is an element of a kind of Lego, in which integers (1.11b,e) generate sequences with corresponding values.The Lego made from (1.11) is shown in Fig. 2. Here, arrows ↔, ↕ show transitions Arrows ⇒⇒⇒ show transitions according to rule (1.9).Values of numbers in square brackets [θ] are multiples of three.
Arrows − show the beginning of the trajectory of the sequence of Example 3. [1].This is a periodically growing oscillating sequence.It is built on the basis of the reverse algorithm (Lemma 3.1).However, sequences with multiples of three indices do not generate other sequences Table 1, which forced the author to postulate the equivalence of sequences of the type [θ] to sequences with indices 4[θ] + 1 and to apply rule (1.9) to implement transitions between them.

Conclusion
The periodically oscillating increasing sequence of Example 3.7 [1] is not related to the well-known Collatz problem [2], and does not indicate failure of the Collatz hypothesis.So, the book "Collatz Hypothesis" is not closed, and the story continues.