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The Proof of the 3X + 1 Conjecture
Lanzhou Institute of Technology, Lanzhou, China
College of Mathematics and Information Science, Northwest Normal University, Lanzhou, China
School of Cyberspace Security, Gansu Politics and Law College, Lanzhou, China
Beijing Forestry University, Beijing, China
- 1 Lanzhou Institute of Technology, Lanzhou, China
- 2 College of Mathematics and Information Science, Northwest Normal University, Lanzhou, China
- 3 School of Cyberspace Security, Gansu Politics and Law College, Lanzhou, China
- 4 Beijing Forestry University, Beijing, China
Advances in Pure Mathematics·Volume 12 (2022)·Pages 10–28·Published 25 January 2022·DOI10.4236/apm.2022.121002
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Abstract
In this paper, we use two new effective tools and ingenious methods to prove the 3X + 1 conjecture. By using the recursive method, we firstly prove that any positive integer can be turned into an element of fourth column of the infinite-row-six-column-matrix after a finite times operation, thus we convert “the 3X + 1 conjecture” into an equivalent conjecture, which is: Any positive integer n must become 1 after finite operations under formation of σ( n ) , where Then, with the help of the infinite-row-four-column-matrix, we continue to use the recursive method to prove this conjecture strictly.
KeywordsThe 3X + 1 Conjecture(Z<sup>+</sup>)<sub>&#8734x6</sub>(Z<sup>+</sup>)<sub>&#8734x4</sub>Transformation
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