Turkish Journal of Analysis and Number Theory
ISSN (Print): 2333-1100 ISSN (Online): 2333-1232 Website: http://www.sciepub.com/journal/tjant
Open Access
Journal Browser
Go
Turkish Journal of Analysis and Number Theory. 2020, 8(2), 49-51
DOI: 10.12691/tjant-8-2-5
Open AccessArticle

The Collatz Conjecture and Linear Indefinite Equation

Li Jiang1,

1Beijing, China

Pub. Date: July 19, 2020

Cite this paper:
Li Jiang. The Collatz Conjecture and Linear Indefinite Equation. Turkish Journal of Analysis and Number Theory. 2020; 8(2):49-51. doi: 10.12691/tjant-8-2-5

Abstract

For the collatz conjecture, we define an iterative formula of odd integers according to the basic theorem of arithmetic, and give the concept of iterative exponent. On this basis, a continuous iterative general formula for odd numbers is derived. With the formula, the equation of cyclic iteration is deduced and get the result of the equation without a positive integer solution except 1. On the other hand, the general formula can be converted to linear indefinite equation. The solution process of this equation reveals that odd numbers are impossible to tend to infinity through iterative operations. Extending the result to even numbers, it can be determined that all positive integers can return 1 by a limited number iterations.

Keywords:
Collatz conjecture 3x + 1 problem Syracuse problem iteration

Creative CommonsThis work is licensed under a Creative Commons Attribution 4.0 International License. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/

References:

[1]  J C Lagarias, The 3x+1 problem and its generalizations. Amer. Math. Monthly, 1985(92): 3-23.
 
[2]  I Krasikov, J C Lagarias. (2003). Bounds for the 3x+1 problem using difference inequalities. Acta Arithmetica, 109(3): 237-258.
 
[3]  Dengguo FENG, Xiubin FAN, Liping DING, Zhangyi WANG. On the nonexistence of non-trivial small cycles of the function in 3x+1 conjecture. Journal of Systems Science and Complexity, 2012, 25: 1215-1222.
 
[4]  A Tomas. A non-uniform distribution property of most orbits, in case the 3x + 1 conjecture is true. Journal of Physics A Mathematical and General, 2016, 30(13): 4537-4562.
 
[5]  Changfeng Gao. Application of Number Theory Method in Solving Binary Indefinite Equations. China. Journal of Jinan Vocation College, 2009(72): 89-91.