The 3X + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of the 3X + 1 problem. I...The 3X + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of the 3X + 1 problem. It is worth noting that, both conjectures are infamous for their simplicity in stating but intractability in solving. In this paper, I aim to provide a clear explanation about the reason why these two problems are difficult to handle and have very different characteristics on convergence of the series via creatively applying the probability theory and global expectancy value E(n) of energy contraction index. The corresponding convergence analysis explicitly shows that a = 3 leads to a difficult problem, while a > 3 leads to a divergent series. To the best of my knowledge, this is the first work to point out the difference between these cases. The corresponding results not only propose a new angle to analyze the 3X + 1 problem, but also shed some light on the future research.展开更多
This article introduces a change rule of 3x + 1 problem (Collatz conjecture), it’s named LiKe’s Rule. It’s a map of 3x + 1 problem, and details the path of each step of the change: For any positive integer, change ...This article introduces a change rule of 3x + 1 problem (Collatz conjecture), it’s named LiKe’s Rule. It’s a map of 3x + 1 problem, and details the path of each step of the change: For any positive integer, change by the Collatz conjecture. 1) This positive integer will change to an odd number;2) The odd number must change to a number of LiKe’s second sequence {3<sup>n</sup> – 1| n ∈ Z<sup>+</sup>};3) Then this 3<sup>n</sup> - 1 will change to a smaller 3<sup>n </sup>– 1 and gradually decrease to 8 (that is 3<sup>2</sup> - 1) then back to 1 in the end. If we can determine each step, the Collatz conjecture will be true. This is certainly more valuable than 2<sup>n</sup> (it might even explain 2<sup>n</sup>). And to illustrate the importance of this rule, introduced some important funny corollaries related to it.展开更多
文摘The 3X + 1 problem (Collatz conjecture) has been proposed for many years, however no major breakthrough has been made so far. As we know, the Crandall conjecture is a well-known generalization of the 3X + 1 problem. It is worth noting that, both conjectures are infamous for their simplicity in stating but intractability in solving. In this paper, I aim to provide a clear explanation about the reason why these two problems are difficult to handle and have very different characteristics on convergence of the series via creatively applying the probability theory and global expectancy value E(n) of energy contraction index. The corresponding convergence analysis explicitly shows that a = 3 leads to a difficult problem, while a > 3 leads to a divergent series. To the best of my knowledge, this is the first work to point out the difference between these cases. The corresponding results not only propose a new angle to analyze the 3X + 1 problem, but also shed some light on the future research.
文摘This article introduces a change rule of 3x + 1 problem (Collatz conjecture), it’s named LiKe’s Rule. It’s a map of 3x + 1 problem, and details the path of each step of the change: For any positive integer, change by the Collatz conjecture. 1) This positive integer will change to an odd number;2) The odd number must change to a number of LiKe’s second sequence {3<sup>n</sup> – 1| n ∈ Z<sup>+</sup>};3) Then this 3<sup>n</sup> - 1 will change to a smaller 3<sup>n </sup>– 1 and gradually decrease to 8 (that is 3<sup>2</sup> - 1) then back to 1 in the end. If we can determine each step, the Collatz conjecture will be true. This is certainly more valuable than 2<sup>n</sup> (it might even explain 2<sup>n</sup>). And to illustrate the importance of this rule, introduced some important funny corollaries related to it.
