On the Glide of the 3x+1 Problem

Yuyin YU  Zongxiang YI  Chuanming TANG  Jian GAO  

IEICE TRANSACTIONS on Fundamentals of Electronics, Communications and Computer Sciences   Vol.E102-A   No.3   pp.613-615
Publication Date: 2019/03/01
Online ISSN: 1745-1337
DOI: 10.1587/transfun.E102.A.613
Type of Manuscript: LETTER
Category: Mathematical Systems Science
3x+1,  Collatz problem,  number theory,  

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For any positive integer n, define an iterated function $f(n)=left{ egin{array}{ll} n/2, & mbox{ $n$ even, } 3n+1, & mbox{ $n$ odd. } end{array} ight.$ Suppose k (if it exists) is the lowest number such that fk(n)<n, and the operation of “multiplying by 3 and adding one” occurs O(n) times and that of “dividing by 2” occurs E(n) times from n to fk(n). We conjecture that 2E(n)-1<3O(n)<2E(n). This conjecture is similar to the conjecture proposed by Terras in 1976, and we also give an upper bound for the residual term of n.