A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of x2 over the Count of 3∗x+1 Is Larger than ln3ln2
Author
Source
International Journal of Mathematics and Mathematical Sciences
Issue
Vol. 2020, Issue 2020 (31 Dec. 2020), pp.1-14, 14 p.
Publisher
Hindawi Publishing Corporation
Publication Date
2020-08-25
Country of Publication
Egypt
No. of Pages
14
Main Subjects
Abstract EN
We propose reduced Collatz conjecture and prove that it is equivalent to Collatz conjecture but more primitive due to reduced dynamics.
We study reduced dynamics (that consists of occurred computations from any starting integer to the first integer less than it) because it is the component of original dynamics (from any starting integer to 1).
Reduced dynamics is denoted as a sequence of “I” that represents (3∗x+1)/2 and “O” that represents x/2.
Here, 3∗x+1 and x/2 are combined together because 3∗x+1 is always even and thus followed by x/2.
We discover and prove two key properties on reduced dynamics: (1) Reduced dynamics is invertible.
That is, given reduced dynamics, a residue class that presents such reduced dynamics can be computed directly by our derived formula.
(2) Reduced dynamics can be constructed algorithmically, instead of by computing concrete 3∗x+1 and x/2 step by step.
We discover the sufficient and necessary condition that guarantees a sequence consisting of “I” and “O” to be a reduced dynamics.
Counting from the beginning of a sequence, if and only if the count of x/2 over the count of 3∗x+1 is larger than ln3/ln2, reduced dynamics will be obtained (i.e., current integer will be less than starting integer).
American Psychological Association (APA)
Ren, Wei. 2020. A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of x2 over the Count of 3∗x+1 Is Larger than ln3ln2. International Journal of Mathematics and Mathematical Sciences،Vol. 2020, no. 2020, pp.1-14.
https://search.emarefa.net/detail/BIM-1172667
Modern Language Association (MLA)
Ren, Wei. A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of x2 over the Count of 3∗x+1 Is Larger than ln3ln2. International Journal of Mathematics and Mathematical Sciences No. 2020 (2020), pp.1-14.
https://search.emarefa.net/detail/BIM-1172667
American Medical Association (AMA)
Ren, Wei. A Reduced Collatz Dynamics Maps to a Residue Class, and Its Count of x2 over the Count of 3∗x+1 Is Larger than ln3ln2. International Journal of Mathematics and Mathematical Sciences. 2020. Vol. 2020, no. 2020, pp.1-14.
https://search.emarefa.net/detail/BIM-1172667
Data Type
Journal Articles
Language
English
Notes
Includes bibliographical references
Record ID
BIM-1172667