Exploring properties in reduced Collatz dynamics

We design a computer program that can output reduced dynamics for odd integers with 4k+3, e.g, [3-99999999]. Outputting (reduced) dynamics for much larger integers are also possible. The source code in C is txpo9.c. There are 5 options in arguments for more flexible output. Those data can reveal the properties of reduced dynamics. The most important are ratio and period. We discover the ratio - the count of x/2 over the count of (3*x+1)/2 - is bounded by a constant value, ln2/ln1.5. We prove mathematically that the ratio of reduced dynamics is larger than lambda = ln2/ln1.5 ≈ 0.58496250 formally in another paper. Those data outputted by txpo9.c can be used to verify this bound. The data shows that our analysis on the bound of ratio is right. Indeed, we can give the equation on the count of x/2 and the count of (3*x+1)/2 for any reduced dynamics. That is, CntO(c) = ceil( CntI(c)*lambda), lambda=ln1.5/ln2=0.58469250, c is any reduced dynamics in terms of I and O with length larger than 1. Here, CntO() is a function returns the count of O in c, and CntI() is a function returns the count of I in c.For example, O is a reduced dynamics for even, it is trivial and listed aside. IO is reduced dyancmics for odd with 4k+1, we can check the equation as follows: CntO(IO)=1, CntI(IO)=1, CntO(c) = 1, ceil(CntI(c)*lambda)) = ceil(1*lambda)=ceil(0.58496250)=1.The bound can help us generate all valid reduced dynamics by algorithm, instead of selecting a positive number to compute its reduced dynamics.Besides, we also discover that the period of reduced dynamics exist. That is, if the reduced dynamics of x is a sequence consisting of I and O with length L, then the reduced dynamics of x+2^L equals the reduced dynamics of x. We also formally prove it mathematically in another paper. This period can also observed and verified in the data file outputted by the program txpo9.c.Note that, for the better vision in computer program output, we use “-” to represent I (i.e., (3*x+1)/2) and “0” to represent O (i.e., x/2) .

本团队设计了一款计算机程序，旨在输出奇数4k+3（例如，[3-99999999]）的简化动力学。该程序亦能输出更大整数的简化动力学。C语言编写的源代码文件为txpo9.c。程序提供了五种参数选项，以实现更为灵活的输出。这些数据能够揭示简化动力学的性质，其中尤为关键的是比例和周期。我们发现，比例——即x/2的出现次数与(3*x+1)/2的出现次数之比——被一个常数ln2/ln1.5所界定。我们在另一篇论文中通过数学证明了简化动力学比例大于lambda = ln2/ln1.5 ≈ 0.58496250。由txpo9.c输出的数据可用于验证这一界限。数据显示，我们对比例界限的分析是正确的。实际上，我们可以为任何简化动力学给出x/2和(3*x+1)/2出现次数的方程，即CntO(c) = ceil( CntI(c)*lambda)，其中lambda=ln1.5/ln2=0.58469250，c是以I和O表示的任意长度大于1的简化动力学。在此，CntO()函数返回c中O的出现次数，CntI()函数返回c中I的出现次数。例如，O代表偶数的简化动力学，是显而易见的，故在此单独列出。IO代表奇数4k+1的简化动力学，我们可以通过以下方程进行验证：CntO(IO)=1，CntI(IO)=1，CntO(c) = 1，ceil(CntI(c)*lambda)) = ceil(1*lambda)=ceil(0.58496250)=1。这一界限有助于我们通过算法生成所有有效的简化动力学，而非仅通过选择一个正数来计算其简化动力学。此外，我们还发现简化动力学的周期性存在。即，如果x的简化动力学是一个由I和O组成的序列，其长度为L，那么x+2^L的简化动力学与x的简化动力学相同。我们也在另一篇论文中通过数学形式化地证明了这一点。这一周期性可以在程序txpo9.c输出的数据文件中观察到并得到验证。请注意，为了在计算机程序输出中提供更好的视觉效果，我们使用“-”来表示I（即(3*x+1)/2），而使用“0”来表示O（即x/2）。