ON THE NONEXISTENCE OF NONTRIVIAL SMALL CYCLES OF THE μ FUNCTION IN 3x+ 1 CONJECTURE  

作　　者：Dengguo FENG Xiubin FAN Liping DING Zhangyi WANG 

机构地区：[1]State Key Laboratory of Information Security,Institute of Software,Chinese Academy of Sciences [2]Institute of Software,Chinese Academy of Sciences

出　　处：《Journal of Systems Science & Complexity》2012年第6期1215-1222,共8页系统科学与复杂性学报（英文版）

基　　金：supported by Natural Science Foundation of China under Grant Nos.60833008 and 60902024

摘　　要：This paper studies the property of the recursive sequences in the 3x ＋ 1 conjecture. The authors introduce the concept of μ function, with which the 3x ＋ 1 conjecture can be transformed into two other conjectures： one is eventually periodic conjecture of the μ function and the other is periodic point conjecture. The authors prove that the 3x ＋ 1 conjecture is equivalent to the two conjectures above. In 2007, J. L. Simons proved the non-existence of nontrivial 2-cycle for the T function. In this paper, the authors prove that the μ function has nol-periodic points for 2 ≤ 1 ≤12. In 2005, J. L. Simons and B. M. M de Weger proved that there is no nontrivial/-cycle for the T function for 1 ≤68, and in this paper, the authors prove that there is no nontrivial l-cycle for the μ function for 2 ≤ 1≤ 102.

关 键 词：Diophantine equation eventual period periodic point 3x ＋ 1 conjecture. 

分 类 号：O156[理学—数学]