The Collatz Conjecture and Linear Indefinite Equation

Li Jiang^1,

¹Beijing, China

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

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