New Method to Solving the Collatz Conjecture 3 with High-Prize Money

Chapter 2: Derivation of CS -Vibration and CS-Plot (article 3)

   Please read "Introduction to a New Method in Chapter 1 (article 1)"
before reading Chapter 2(article 3).

   Chapter 1 also posts links to each chapter.

   When translating from Japanese to English, subtle nuances of the original articles may not be fully conveyed. If you would like to view the original Japanese articles, please see Chapter 1 of Japanese Article 1.

   Summary Video of the New Method in English
　

2.3 Relationship between the Derivation of the CS-linear-equations and CS-Plot Points 

2.3.1 Before Explanation

   Section 2.2 presents the CS plot and the CS-linear-equations using the odd integer $${No=27}$$ as the initial value. It is found that the 41 points on the CS-plot are located "close" to the CS-linear-formula.
(When the initial value is 27, only one point on the x-axis,$${(1/4,0)}$$, is away from the CS-linear-formula.)

   Do these properties hold for any odd number $${No}$$?

Yes, they do.

   This proof is considered key, therefore it has been presented as a separate article in Section 2.3 of Chapter 2, as mentioned at the end of Section 2.1.

   The following demonstrates the derivation of the CS-linear-equations and proves that the CS-plot points exist "near" the CS-linear-equations for any odd integer $${No}$$. The definition of "near" is given in the first step.

   A method that "completely" coincides will be explained in order from Chapter 3 onwards.
   It will not only explain the method that simply coincides, but also explain that the relationship between two equations in the method inherently includes the unique characteristics of the Collatz conjecture's number sequence. (For a specific example of a complete coincidence diagram, see Chapter 4, Figure 4-1.)

   If you find the content of this article to be "I see" , please consider giving it a good rating. Your feedback will motivate me to continue my analysis in the future.

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Variables, Definitions, and Functions up to Section 2.3 in Chapter 2:

$${N}$$, $${No}$$,
total number of steps, step number, CS-vibration, C-transformation,
CS-plot, CS-linear-equations, 3C+1 transformation,
$${c(No)}$$, $${z_{s}}$$, $${f_{c}(x)}$$, $${f_{s}(x)}$$, distance$${D}$$、

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   The new method is introduced in Articles 1-10 and Chapters 1-8.
The translation from Japanese to English has been completed for all articles, both free and paid. (April 2024)

2.3.2 Preparation for Derivation