论文标题
在广义的Collatz序列的周期上
On Cycles of Generalized Collatz Sequences
论文作者
论文摘要
我们探索了广义Collatz序列的周期和收敛，其中$ 3N+1 $的原始Collatz功能被$ 3N+K $取代。我们提出了GC循环的生成函数，并在此类序列中显示了循环的特定遗传结构。循环结构在这种继承之间是不变的，并且比循环元素看起来更基本。结果是在某些序列中可能有大量的周期。 GC也可以看作是整数空间分区函数，并且这些分区以及Collatz图在序列之间继承。还提出了GCS循环与某些指数二氧剂方程之间的有趣联系。
We explore the cycles and convergence of Generalized Collatz Sequence, where $3n+1$ in original collatz function is replaced with $3n+k$. We present a generating function for cycles of GCS and show a particular inheritance structure of cycles across such sequences. The cycle structure is invariant across such inheritance and appears more fundamental than cycle elements. A consequence is that there can be arbitrarily large number of cycles in some sequences. GCS can also be seen as an integer space partition function and such partitions along with collatz graphs are inherited across sequences. An interesting connection between cycles of GCS and certain exponential Diophantine equations is also presented.
