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Feng, D., Fan, X., Ding, L. and Wang, Z. (2012) On the Nonexistence of Nontrivial Small Cycles of the μ Function in 3x + 1 Conjecture. Science and Complexity, 25, 1215-1222.
https://doi.org/10.1007/s11424-012-0280-5

has been cited by the following article:

• TITLE: The Proof of the 3X + 1 Conjecture

  AUTHORS: Maoze Wang, Yongbao Yang, Zhenxiang He, Meiyi Wang

  KEYWORDS: The 3X + 1 Conjecture, (Z+)∞x6, (Z+)∞x4, Transformation, Module

  JOURNAL NAME: Advances in Pure Mathematics, Vol.12 No.1, January 26, 2022

  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.